Games Conceived

Sum Of Products, Number Of Numbers Game

2011-08-17T13:13:00.001-06:00

Here is a game for 2 players played using an n-by-n grid drawn on paper.

Players take turns. On a turn a player writes any one integer, 1 to n, into any empty square of the grid.

After n^2 total turns, the game is over.

Player 1 gets the sum of the scores for each row. The score for a row is (the number of 1's in the row) * (the number of 2's in the row) * (the number of 3's in the row) *...(the number of m's in the row), where m is the largest integer such that all integers 1 through m occur in that row.

Player 2 gets the sum of the scores for each column. The score for a column is (the number of 1's in the column) * (the number of 2's in the column) * (the number of 3's in the column) *...(the number of m's in the column), where m is the largest integer such that all integers 1 through m occur in that column.

A player gets 1 point for a row/column if there are no 1's in that row/column.

The player with the largest score wins.

And example game:

1 1 2 1 2 5
1 3 2 1 3 5
5 1 1 1 3 1
1 2 1 2 1 2
2 1 1 2 1 1
2 3 5 3 4 1

Player 1 gets:
3*2 + 2*1*2 + 4 + 3*3 + 4*2 + 1*1*2*1*1
=
33 points.

Player 2 gets:
3*2 + 3*1*2 + 3*2 + 3*2*1 + 2*1*2*1 + 3*1
=
31 points.

Player 1 wins.

Thanks,
Leroy Quet

Binary Scramble Game

2011-08-17T13:11:00.000-06:00

Here is a game for any plural number of players.

A list of 0's and 1's is written on a piece of paper by the players taking turns, each player appending a 0 or a 1 onto the right side of the list each turn. After a predetermined number of turns (which is a multiple of the number of players), the first part of this game is over.

In the second part of the game, the players take turns rewriting the entire list each turn with one digit the player chooses flipped from 0 to 1 or from 1 to 0.

The new list cannot match any list previously arrived at during the game.

If the lengths of the runs of 0's and 1's form a permutation of the lengths of the runs from any previous list, then the currently moving player gets a point.

(It doesn't matter if a particular run-length, an element in the permutation, was for a run of 0's or for a run of 1's.)

The game continues until either a player first achieves a predetermined score or until no more moves are possible.

The player with the greatest score wins.

Example game (to start):

001110101 (start: 2,3,1,1,1,1)
000110101 (3,2,1,1,1,1 point)
000111101 (3,4,1,1)
000011101 (4,3,1,1 point)
100011101 (1,3,3,1,1)
101011101 (1,1,1,1,3,1,1)
111011101 (3,1,3,1,1 point)
111010101 (3,1,1,1,1,1,1 point)
111010111 (3,1,1,1,3 point)
... etc.

Thanks,
Leroy Quet

Combining And Dividing Integers Game

2011-07-22T10:47:00.000-06:00

Here is a game for any plural number of players.
(And no grids!)

Players take turns, on each turn writing any positive integer to the end of a growing list of integers. The players do this until there is a predetermined number of integers in the list. (The number of integers is a multiple of the number of players.)

Players then take turns; on each turn a player rewrites the list with one of these changes:
Either:
The player changes one integer in the list to a product of any number of PROPER divisors, each divisor greater than 1, that all multiply to that integer. (The divisors must be placed between the same integers in the list as the integer they replaced, but they can be in any order amongst themselves between those integers.)
Or:
The player replaces two adjacent integers in the list with their sum. (The sum goes in the same location within the list as the two integers it replaces.)

The first player to get a list completely of primes is the winner.

If a large predetermined number of moves have taken place, and no one has yet won, then the game is a tie.

Example:

2 players:

List to start:
2, 4, 10, 4, 1, 10
P1:
2, 4, 2, 5, 4, 1, 10
P2:
2, 4, 2, 5, 5, 10
P1:
2, 6, 5, 5, 10
P2:
2, 6, 5, 15
P1:
2, 6, 20
P2:
8, 20
P1:
8, 4, 5
P2:
12, 5
P1:
2, 3, 2, 5

Player 1 wins.

Thanks,
Leroy Quet

Divisor Over Sums Game

2011-06-12T05:59:00.000-06:00

Here is another game that uses an n-by-n grid drawn on paper.

This game is for 2 players.

Players take turns placing 1,2,3,... n^2 into empty squares of the grid, one number each turn.
(So, player 1 places the odd numbers into the grid, and player 2 places the evens.)

Let H represent the collection of variables h, where each h is the sum of a HORIZONTALLY-adjacent pair of integers in the grid.

Let V represent the collection of variables v, where each v is the sum of a VERTICALLY-adjacent pair of integers in the grid.

(So, H and V each consist of n*(n-1) variables.)

Player 1 then chooses a variable d which divides at least one of the h's.
Player 1 gets as a score:
d*(number of h's that are divisible by d)

Player 2 then chooses a variable d' which divides at least one of the v's.
Player 2 gets as a score:
d'*(number of v's that are divisible by d')

The player with the largest score wins.

For example, let us say two poorly-playing players have this n=4 grid:

08 09 01 10
03 11 16 07
02 14 15 05
04 12 06 13

H =
(17,10,11, 14,27,23,
16,29,20, 16,18,19)

Let us say that player 1 picks 4, then he gets 4*3 = 12 points.
(Because h's = 16, 20, and 16 are divisible by 4.)
Had he picked 29, he could have gotten 29, however.

V =
(11,5,6, 20,25,26,
17,31,21, 17,12,18)

Player 2 picks 3 and gets 3*4 = 12 points.
She should have picked 31.

Yes, you can always score at least n*(n-1) by picking d or d' = 1.

As implied in the example, a player is responsible for finding his/her own score.

Thanks,
Leroy Quet

Palindromic Rows/Columns

2011-06-04T15:00:00.001-06:00

This game is for 2 players.

Start by drawing an n-by-n grid on paper.
(Who would have guessed??)

The players take turns filling in empty squares of the grid, one square per move. Each player fills in floor(n^2/4) squares, so that there are a total of 2*floor(n^2/4) squares (about 1/2 of grid) filled in at game's end.

Player 1 gets as a score the number of grid-squares whose states need to be changed (from filled to unfilled, or vice versa) in order to make each ROW a palindrome.

Player 2 gets as a score the number of grid-squares whose states need to be changed in order to make each COLUMN a palindrome.

The player with the SMALLEST score wins.

{I know that the rules could have had Player 1 score with the number of squares needed to make the *columns* into palindromes, and Player 2 could have had the rows, then the *highest* score wins; but if each player attempts to determine their own score, then with my way each player has an incentive to be as efficient as possible in determining how many squares need to be changed to achieve the palindromes.}

Thanks,
Leroy Quet

(Not)Divisors Or (Not)Multiples

2011-05-12T05:40:00.000-06:00

This game is for any plural number of players.

Start with an n-by-n grid drawn on paper. (I suggest an n of at least 4 or 5 if there are two players who are beginners. I suggest a larger n if there are more players.)

The game starts with Player 1 placing any integer >=2 in any square of the grid.
Thereafter, the players take turns. On a turn a player places any integer >= 2 in any empty square that is adjacent to and above, below, left of, or right of any square with an integer in it already.

Each number must not have been used in the game yet.

And each number written down must be a divisor or a multiple of each preexisting number that is orthogonally (above, below, left of, right of) adjacent to it.

And each number written down must NOT be a divisor or a multiple of (may be coprime or not coprime to) each preexisting number that is diagonally adjacent to it.

The last player able to move wins.

Example finished game:
(n = 4. ** denotes an empty square. Leading 0's here are for formatting.)

** ** ** 15
** ** 10 05
08 04 02 **
** ** 06 03

Thanks,
Leroy Quet

Plexus Via Midpoints And Endpoints

2011-05-07T06:57:00.001-06:00

This is a game for two players.

Start with a lattice of n-by-n dots (which are the vertices of an (n-1)-by-(n-1)-squares grid). n is odd.

One of the players draw a straight line-segment from the middle dot of the lattice to any adjacent dot above, below, left of, or right of the central dot.
(It doesn't matter which direction this line-segment is drawn.)

The game consists of "full-moves", each of which consists of two "half-moves". In a full-move, one player moves then the other. The order of the players in the full-moves alternates.
So, we have the players moving like this:
(1,2),(2,1),(1,2),(2,1),(1,2),(2,1),...

In a full-move, the players each draw a straight line-segment from a dot already connected with another dot via a line-segment, to an adjacent dot (in the direction of either above, below, left of, or right of). The adjacent dot must have no line-segments yet connected with it.

The first player to move within a full-move decides if his/her line segment will be drawn from a dot already connected to 2 or 3 line-segments (a midpoint), or from a dot already connected to by exactly one line-segment (an endpoint).

The second player to move in the full-move must also draw his/her line-segment from a midpoint or from an endpoint just as the first player in the full-move did.
In other words, if the first player in the full-move draws from a midpoint, the second player must also draw from a midpoint. And if the first player drew from an endpoint, the second player must also draw from an endpoint.
(A player is free to decide which midpoint or endpoint to draw from, however, provided that the dot he/she draws to doesn't have any line segments connecting to it yet.)

If it is impossible for the second player in the full-move to do as the first player did (draw from midpoint or draw from endpoint), he/she loses his turn in this particular full-move, and the first player to move in that full-move then gets a point. (The second player still moves first in the next full-move, though.)

The game continues until all dots have each been connected with at least one line-segment.

The player with the largest score wins.

Thanks,
Leroy Quet

Palindromes/Antipalindromes In Sequence

2011-04-26T05:51:00.000-06:00

Game for 2 players.

Make a row of m squares drawn on paper, where m is decided by both players.
(I suggest that m be >= 16 for beginners.)

Players take turns. On a turn, a player places either a 0 or a 1 into any empty square.

After a total of m moves, and when each square has one digit in it, play is over.

Player 1 gets the sum of the lengths of all (not necessarily distinct but possibly overlapping*) palindromes
(a(1),a(2),a(3)...a(3),a(2),a(1))
of EVEN length (in the row of 0's and 1's) as her/his score.
Player 1's score =
sum{k=1 to [m/2]} 2k*(# of palindromes of length 2k).

Player 2 gets the sum of the lengths of all (not necessarily distinct but possibly overlapping*) antipalindromes
(a(1),a(2),a(3)...1-a(3),1-a(2),1-a(1))
(of even length) (in the row of 0's and 1's) as her/his score.
Player 2's score =
sum{k=1 to [m/2]} 2k*(# of antipalindromes of length 2k).

*(By "not necessarily distinct", I mean the patterns of 0's and 1's in different palindomes/antipalindromes may match. I of course don't mean the SAME palindrome/antipalindrome {in the same position and of the same length} can count more than once.)

The player with the largest score is the winner.

For example, let us say we have the following row (m=16):
1001001101000100

We have the following palindromes of even length:
00, 00, 11, 00, 00, 00
1001, 1001, 0110

So player 1 gets (2*6 + 4*3 =) 24 points.

We have the following antipalindromes:
10, 01, 10, 01, 10, 01, 10, 01, 10
0011, 1010
100110, 110100
01001101

So player 2 gets (2*9 + 4*2 + 6*2 + 1*8=) 46 points.

Player 2 wins.

(Did I make a mistake counting up the palindromes and antipalindromes in my example?)

I suggest, if this game isn't played on a computer, that players count their own (anti)palindromes, and confirm them with their opponent. (If a player misses any that would contribute to his/her own score, it would be his/her own fault.)

Thanks,
Leroy Quet

Divisors/Multiples Sequence Game

2011-04-22T12:25:00.000-06:00

Here is a game for any plural number of players.

r is a positive integer agreed upon by all players before the start of the game.
I suggest that r be congruent to 1 mod {the number of players}.

The game starts with a(1) = 1 and b(1) = 1. m and n both equal 1.

Players alternate moves.

(*)n=n+1.
(**)m=m+1.
A player on his move picks an integer a(m) that is either a divisor or a multiple of a(m-1).
a(m) must be <= r.
The player gets max(a(m),a(m-1))/min(a(m),a(m-1)) added to his score if a(m) is not among (b(1),b(2),...b(n-1)).
But the player gets
2*max(a(m),a(m-1))/min(a(m),a(m-1))
added to his score if a(m) is among (b(1),b(2),...b(n-1)).
The player continues his turn as far as he wants, but until a(m) is not amongst (b(1),b(2),...b(n-1)).
If the player is to continue his move, he goes to (**).
If a(m) is not among (b(1),b(2),...b(n-1)) and the player wants to end his move, then b(n) = a(m), and switch whose turn it is, go to (*) if n is < r.

Play until n = r, and (b(1),b(2),...b(n)) is a permutation of the integers 1 through r.
The winner is then the player with the LOWEST score.

Thanks,
Leroy Quet

Sums Equal 1,2,3,...m In Small Grid

2011-04-09T10:01:00.002-06:00

This can be considered both a puzzle and a solitaire game.

Make an n-by-n grid on paper. (I suggest that n = 3 for beginners.)

Fill the grid with UNIQUE positive integers, one integer per square of the grid. The numbers need not be consecutively valued necessarily.

Your score is the largest integer m such that integers 1 through m all occur as sums within the grid (without missing any positive integers <=m).

A "sum" is of any number of addends (possibly just 1) that are all *consecutively placed* within a row of the grid or a column of the grid. (No diagonals in this variation.)

So, for example, if we have the following 3-by-3 grid:

2 9 8
3 1 10
7 6 15

...the sums 1 through 19 all occur in this grid. So, I get a score of 19.
(Notice that some values of sums occur more than once.)

I bet it is easy to do better than I did with a 3-by-3 grid, since my grid is inefficient, and I didn't try too hard to find it.

(Note: There are 27 possible sums in a 3-by-3 grid of 1, 2, or 3 consecutively placed addends. I don't know what the largest possible score is for a 3-by-3 grid, though.)

You can "play" someone else by both of you trying to score as well as you can on same-sized grids. Just try to outscore your opponent.

Update: The largest score I personally received on the 3-by-3 grid is 20. But someone using a computer, I think, found a 3-by-3 grid with a score of 25.
Update2: Someone else found a 3-by-3 grid with a score of 26.

Thanks,
Leroy Quet

Multiples/Divisors Blob Game

2011-04-01T11:42:00.001-06:00

This game is for 2 players.

Start with an n-by-n grid drawn on paper. (n should be at least 8, I suggest.)

The first player fills in any one of the grid's squares to start.

Thereafter, players continue to take turns each filling in one empty square each turn.
After the first move, each square that is filled in must be immediately next to (in the direction of above, below, left of, or right of) at least one square that is already filled in.

After both players have each filled in floor(n^2/4) squares (for 2*floor(n^2/4) squares filled in total), the game is over.

Player 1 gets as a score the number of ROWS of the grid meeting this condition: Every run-length (of runs each of either all filled in squares or all empty squares) in that particular row is either a multiple or divisor of every other run length in that row.

Player 2 gets as a score the number of COLUMNS of the grid meeting this condition: Every run-length (of runs each of either all filled in squares or all empty squares) in that particular column is either a multiple or divisor of every other run length in that column.

The player with the largest score wins.

Example: (n=12)

o o * o o o o o o o o o
o o * * o o * * o o o o
o o o * * o * * o * * *
o o o o * * * * * * o *
o o o * * * * o o * * *
o o o * o o * o * * * o
o * * * * * * * o o * o
o * o o * * o * * o * *
* * o o * * * * o o * *
* o o o * o o * * o * o
* * o * * * o o o * * *
o o o o o * * o * * * o

Run-lengths of rows:
(2,1,9)
(2,2,2,2,4) Point!
(3,2,1,2,1,3)
(4,6,1,1)
(3,4,2,3)
(3,1,2,1,1,3,1)
(1,7,2,1,1)
(1,1,2,2,1,2,1,2) Point!
(2,2,4,2,2) Point!
(1,3,1,2,2,1,1,1)
(2,1,3,3,3)
(5,2,1,3,1)

Run-lengths of columns:
(8,3,1)
(6,3,1,1,1) Point!
(2,4,1,5)
(1,2,1,3,3,1,1)
(2,3,1,5,1)
(3,2,1,3,1,2)
(1,6,1,1,2,1) Point!
(1,3,2,4,2)
(3,1,1,1,1,1,1,1,1,1) Point!
(2,4,4,2) Point!
(2,1,1,8) Point!
(2,3,2,2,1,1,1)

Player 2 wins, 5 to 3.

Thanks,
Leroy Quet

Sum Kind Of Game

2011-03-21T11:32:00.000-06:00

This is a game for two players.
Start with an n-by-n grid drawn on paper, where n is even.

Players take turns placing numbers into the empty grid squares, one number per move.
Player 1 places into the grid a 1 then a 2 then 3 then... then the value of n^2/2-1 then finally the value of n^2/2.
Player 2, on the other hand, goes the opposite way, placing into the grid the value of n^2/2 then the value of n^2/2-1 then... then 3 then a 2 and finally a 1.

Player 1 gets as a score the number of squares with integers S where each is the sum of the values in two different squares of the same ROW (same row as their sum S).

Player 2 gets as a score the number of squares with integers S where each is the sum of the values in two different squares of the same COLUMN (same column as their sum S).

(The addends can be the same value, but must be in different squares.)

It doesn't matter who wrote down each number as far as scoring is concerned.

Largest score wins.

Sample game: (n=6)

01 02 03 04 05 18
12 17 05 15 11 10
10 14 13 17 03 06
09 14 06 04 09 02
08 16 07 15 08 12
16 18 07 13 11 01

Player 1 gets for each row (top to bottom):
3, 2, 2, 1, 2, 1
11 points.

Player 2 gets for each column (left to right):
2, 2, 1, 1, 3, 2
11 points.

(I may have made a mistake or two figuring these out.)

A tie, if I didn't err.

----

Is there a bias in favor of one player in the game because strategies are different due to the two orders the numbers are placed into the grid?

Thanks,
Leroy Quet

Flipping Bits/Squares

2011-03-12T12:28:00.001-07:00

This game is for any plural number of players.

You need a blank piece of paper and a pen/pencil.

A predetermined number of rounds are played.

Start each round by making a row of n squares, where n is at least 7 or more if there are 2 players; n is larger if there are more players.
Fill in every other square.

Take turns.

On a turn a player can change any ONE square from filled in to not filled in, or vice versa.
Make a new row of squares to reflect the move.

A move may not lead to a pattern of how the squares are filled that has already existed in the round.
If all patterns achievable by changing one square (flipping one bit) lead to patterns that have already existed in the round, then a player flips two bits. If all patterns achievable by flipping two bits also lead to patterns that already occurred in the round, then flip three bits; etc.
A player MUST move with the fewest number of bits flipped as possible that will lead to a new pattern for the round.

As soon as a player achieves with their move a pattern with exactly one bit/square a different color (either filled or not filled) than the rest of the row, then that player gets {the number of squares from the left side of the row that is this unique square's position} added to his/her score.

The round is then over.
(So, one player scores each round.)

Play a predetermined number of rounds, adding up each player's scores.

The player with the largest grand total wins.

Example round: (n = 7)

(*)( )(*)( )(*)( )(*): start
(*)( )( )( )(*)( )(*): player 1
(*)( )( )( )(*)( )( ): player 2
(*)( )( )(*)(*)( )( ): player 1
(*)(*)( )(*)(*)( )( ): player 2
(*)(*)( )(*)(*)( )(*): player 1
(*)(*)(*)(*)(*)( )(*): player 2
Player 2 gets 6 points, since the blank square is the 6th square from the left.

(I wonder what the funnest n is for any given number of players, especially for 2 players.)

Thanks,
Leroy Quet

Another Binary Primes Game

2011-02-22T10:48:00.001-07:00

This is a game for 2 players.

Draw a single column of n squares.

Players take turns. On a turn a player writes an x into any blank square. Each player makes a total of floor(n/4) x's, so about half the squares are filled at game's end.

A player's score equals the number of DISTINCT primes they can find whose binary representation is a substring in the columns, if each x is interpreted as a 1 and each blank square is interpreted as a 0. Player 1 finds binary primes written from top to bottom in the column, and player 2 finds primes written from bottom to top in the column.

Largest score wins.

Thanks,
Leroy Quet

Binary Primes Game

2011-01-15T06:32:00.001-07:00

This is a game for two players.

Start by drawing an n-by-n grid on paper. (I don't suggest that n be too big, unless you are playing this game on a computer.)
n is odd. (Thanks to Ilmari Karonen for pointing out that a win can always be forced by player 2 if n is even {except for a likely small number of even exceptions}.)

The players take turn filling in squares, each player filling in one empty square per move.
Each player fills in floor(n^2/4) squares total.
(So, 2*floor(n^2/4) squares are filled in all together at the game's completion.)

Now, interpret the filled-in (black) squares and the blank (white) squares in each row and column as the digits of a binary number. In any particular row or column, either all the black squares represent a 0 or all of the black squares represent 1. And the white squares each equal the opposite binary digit than the black squares represent in that row/column. And the binary number can be read either top to bottom, or bottom to top (for each column), or left to right, or right to left (for each row).

So, to be clear, the binary digit (0 or 1) represented by a color has to remain the same within any particular row or column, but can differ between different rows and columns.

And the direction the binary number is read can differ between different rows and columns.

So, after the game is complete, the players go through and write down in two lists, one for the columns and one for the rows, the decimal representation of the largest possible PRIME possible, if any is possible, for each row and each column.

Player 1 gets as a score the product of all of the primes in the columns.
Player 2 gets as a score the product of all of the primes in the rows.

The player with the largest score wins.

Example (randomly "played" with no strategy):
n=6:

. * * . . *
* * . . . *
. * * * . .
. * * . . .
* . * * * .
. * . . * *

Largest primes, or 1 if all possibilities are composite or 1:

For columns:
1, 61, 29, 53, 3, 1.
Score =
61*29*53*3 = 281271.

For rows:
1, 1, 1, 1, 29, 19.
29*19 = 551.

Player 1 (columns) wins, obviously.

Thanks,
Leroy Quet

Permudrome -- Grid Game

2010-12-21T04:45:00.001-07:00

Here is a game for 2 players.
The game's name is a combination of the words "permutation" and "palindrome".

Start with an n-by-n grid,
where n is a multiple of 4.
I suggest that n is >= 8.

The players take turns. On a turn a player draws two x's into the grid, each x into an empty square such that no column or row has more than one x.

After there is exactly one x in each row and in each column -- n x's total, n/4 moves for each player -- play is over.

Write down the (n-1) absolute values in order, of the changes in the vertical positions of adjacent x's from column to column, along the bottom of the grid.
Write down the (n-1) absolute values in order, of the changes in the horizontal positions of adjacent x's from row to row, along the left side of the grid.

Player 1 gets as a score the length of the largest palindromic subsequence within the sequence of vertical changes written along the bottom of the grid.

Player 2 gets as a score the length of the largest palindromic subsequence within the sequence of horizontal changes written along the left side of the grid.

Largest score wins. (Ties are possible.)

Example: n=12:

. . x . . . . . . . . .
. . . . x . . . . . . .
. . . . . . x . . . . .
x . . . . . . . . . . .
. . . . . x . . . . . .
. . . . . . . x . . . .
. x . . . . . . . . . .
. . . x . . . . . . . .
. . . . . . . . x . . .
. . . . . . . . . . . x
. . . . . . . . . . x .
. . . . . . . . . x . .

Changes in vertical positions column to column:
3,6,7,6,3,2,3,3,3,1,1
The largest palindromic subsequence is (3,6,7,6,3). Player 1 gets 5 points.

Changes in horizontal positions row to row:
2,2,6,5,2,6,2,5,3,1,1,
The largest palindromic subsequence is (5,2,6,2,5). Player 2 gets 5 points.

It is a tie.

What about strategies for this game?

Thanks,
Leroy Quet

One x Twice

2010-12-13T05:48:00.002-07:00

For 2 players.
Start with an n-by-n grid drawn on paper.

A move consists of both players each secretly picking an integer between 1 and n.
Both numbers are then revealed. An x is then drawn in the grid-square that has the column number of player 1's number, and has the row number of player 2's number.
So, in other words, player 1 picks the horizontal position of the number, and player 2 picks the vertical position.

If the x lands in an empty square, then the game continues.

But, however, the first time an x lands in a square that already has an x, then the game is over. Player 1 wins if this final x was written on an oddly numbered move. Player 2 wins if this x was written on an evenly numbered move.
So, in other words, if there are an odd number of x's at game's end -- and an even number of squares with x's -- then player 1 wins. If there are an even number of x's -- and an odd number of squares with x's -- then player 2 wins.

What kind of strategies will help you win at this game (if you cannot read the other player's mind)?

Thanks,
Leroy Quet

Line & Unobscured Dots Game

2010-12-13T05:48:00.001-07:00

Here is an unoriginal game for 2 players.

You need a blank piece of paper and a pen/pencil, maybe 2 pens/pencils of different colors.

To start, someone draw a dot in the middle of the piece of paper. Then each player draws a different dot on the paper. (So, you have 3 dots.)

Thereafter, the players take turns forming a line of connected straight line-segments on the paper, plus drawing dots. On their turn, a player draws a straight line segment from THEIR END (of their color, if the players are using differently colored pens/pencils) of the connected string of line-segments (or from the central dot if this is the player's first time drawing a line-segment during the game) to any undrawn-to dot (a dot without a line-segment connected to it), such that the line-segment doesn't pass through any other line-segments or through any other dots along the way.

Then, on the same move, the player draws 2 dots, neither on a line or on another dot. One dot is "visible" by the player's own end of the line of connected line-segments. The other dot is visible by the other player's end of the line of connected line-segments.

By 2 points being "visible" to each other, it is meant that it is possible to draw a straight line-segment between the two points, and that line-segment doesn't pass through any intervening dots or lines.

After a fixed number of moves, the same number of moves for both players, the game is over.

The winner has, at game's end, the most number of undrawn-to dots visible from their end-point of the line of connected line-segments.

Note: If playing with 2 differently colored pens/pencils, it doesn't matter what color the dots are. The only reason for using 2 different colors is to make it easier to see whose end of the line of connected line-segments is whose.

Thanks,
Leroy Quet

x's and lines

2010-11-27T06:24:00.000-07:00

(Is this game original? I myself may have come up with something similar earlier.)

This is a game for 2 players.

You have an n-by-n grid drawn on paper (as almost always).
n should be >= 8, I suggest.

In the first part of the game, the players take turns placing a total of n x's into the grid, where one x is placed in an empty square of the grid each move.

In the second part of the game, players take turns. On a turn, a player draws a line (through the centers of the intervening squares) either up, right, down, or left from the last x drawn to by the other player. The (maybe bending) line may take at most one right-angle turn. And it must end at an x not drawn to/from yet. (Player 1 draws from any of the x's on her/his first move.) (What I mean by "the line may take at most one right angle turn" is that the x's will either be connected by a single straight line-segment {if both x's are in the same row or column} or they will be connected by two perpendicular, connected straight line-segments {if the x's are in both a different row and a different column}).

Lines can't pass through a line already drawn in the second part of the game. And the line cannot pass through an x on its way between two other x's during a move. The line may, though, share a corner with, or coincide partially with (both line-segments horizontal or both line-segments vertical), a line drawn previously during the second part of the game.

A player must move if it is possible.

The last player able to move LOSES.

(Note: Unlike some other games I have posted recently, more than one x or no x's at all may be in any row or column.)

Thanks,
Leroy Quet

Multiplications Within The Permutation

2010-11-09T05:16:00.001-07:00

This is a game for any plural number of players. Let the number of players be m.

Start with an n-by-n grid drawn on paper, where n = k*m + 1, k is some integer >= 3.

In the first part of the game, players take turns placing x's in empty squares of the grid, one x per turn, such that no more than one x is in each row and in each column of the grid.

After exactly n x's are placed in the grid, the first part of the game is over.

The second part of the game starts with a 1 being placed in the leftmost square with an x in it. Players then take turns.
On the jth move (starting at move # 1) of the second part of the game, the moving player places a (j+1) in any square with an x and without a number already in it.
He/she gets added to her/his score:
|x(j)-x(j+1)| * |y(j)-y(j+1)|,
where x(j) is the number of squares from the bottom of the grid where the square with the j in it is located, and y(j) is the number of squares from the left side of the grid where the square with the j in it is located.

So, what we are adding to the moving player's score (the score of the player writing a j+1 in a square) is the product of {the change in horizontal distance between the squares with j and j+1 in them} and {the change of vertical distance between the squares with j and j+1 in them}.

When the nth x is numbered with a n= m*k+1, the game is over.

Largest score wins.

Here is an example:
n=7. m =2.
. 3 . . . . .
. . 6 . . . .
. . . . . . 5
. . . . 7 . .
1 . . . . . .
. . . 4 . . .
. . . . . 2 .

Squares 1 to 2: 5*2 = 10
Squares 2 to 3: 4*6 = 24
Squares 3 to 4: 2*5 = 10
Squares 4 to 5: 3*3 = 9
Squares 5 to 6: 4*1 = 4
Squares 6 to 7: 2*2 = 4

Player 1 gets: 10+10+4 = 24 points.
Player 2 gets: 24 + 9 + 4 = 37 points.
Player 2 wins.

Thanks,
Leroy Quet

Prime Target Game

2010-10-30T07:34:00.000-06:00

A game for any plural number of players: (Number of players = m.)

Draw m*k+1 incrementally larger concentric circles on a piece of paper, where k is some positive integer >= 2.

Subdivide the circles by drawing n equally spaced rays from their center, where n is at least 6, say.

(So now you should have a target.)

The players start the game by taking turns, and each player on a turn places an integer -- 1 to n and which has not been written down earlier in the game -- in an empty pie-shaped wedge in the central circle. After n moves, there should be a permutation of (1,2,3,...,n) in the central circle, one integer per wedge.

In the second part of the game, the players take turns, each player "completing a ring" on a move. By completing a ring, the player fills in the n sections of the innermost *empty* ring. The player fills in each section of the ring either with the sum of (the integer immediately adjacent to the section, but in the next ring inward) and (the integer one position clockwise to the section, but in the next ring inward), or with the absolute value of the difference between these particular two integers (in the next ring inward).

Example:
\...8..|....../
-\-----|-----/-
..\.2..|..6./
---\---|---/---
8 = 2+6.
(The 8 could have been a 4.)

After the ring is completed, the player gets the number of primes in his latest ring added to his score, OR, if there is exactly one prime in his ring (no more, no fewer), he gets the value of that prime added to his score.

After all rings are completed, the game is over. Largest score wins.

Thanks,
Leroy Quet

Grid Game Of Differences

2010-10-11T13:20:00.000-06:00

This is a game for two player.

Draw an n-by-n grid on paper, where I suggest that n is at least 8.

The players take turns placing x's in the empty squares of the grid, one x per turn.

No two or more x's may be placed in the same row or in the same column of the grid.

After n total moves (when there is exactly one x in each row and column), the game is over.

Now to determine the score:

Reading left to right, write down the (n-1) absolute values of the differences between the consecutive x's' vertical coordinates, in terms of number of squares.

In another list, reading bottom to top, write down the (n-1) absolute values of the differences between the consecutive x's' horizontal coordinates, in terms of number of squares.

Player 1 gets a point for every distinct numerical value occurring in the first list of differences.
Player 2 gets a point for every distinct numerical value occurring in the second list.
If a particular difference occurs at least once in a single list, then the player gets one point for that particular difference.

Largest score wins.

We may need an example here:

n=9:

. x . . . . . . .
. . . . . . . . x
. . x . . . . . .
. . . . . x . . .
x . . . . . . . .
. . . x . . . . .
. . . . . . . x .
. . . . . . x . .
. . . . x . . . .

Player 1's (vertical) differences (reading left to right) are:
4,2,3,3,5,4,1,5
The unique values that occur are:
1,2,3,4,5
Player 1 gets 5 points.

Player 2's (horizontal) differences (reading bottom to top) are:
2,1,4,3,5,3,6,7
The unique values that occur are:
1,2,3,4,5,6,7
Player 2 gets 7 points.

In another variation of this game, count ONLY those differences that occur exactly once (and no more than once).
In this variation, player 1 would have gotten 2 points, for the differences 1 and 2.
Player 2 would have gotten 6 points, for the differences 1,2,4,5,6,7.
(Since 3 is the only difference in this list that occurs more than once.)

Which variation is more fun?

Thanks,
Leroy Quet

Bouncing Pathways Within A Circle: Game

2010-10-06T05:27:00.000-06:00

A game for two players:

First, draw a circle on a piece of paper.

Players start by each drawing a different straight line-segment at any angle they choose from the center of the circle to the circumference.

Players thereafter move like so: (Player 2, Player 1), (Player 1, Player 2), (Player 2, Player 1), (Pl 1, Pl 2), (Pl 2, Pl 1), etc.
So, we have "whole moves", consisting of two moves, with a move by each player. And who moves first in the whole moves alternates.

The first player to move in a whole-move decides if the next line-segment will bounce left or bounce right. This player then draws his straight line-segment in the proper direction (relative to the direction his own last line-segment was traveling) from where his own last line segment ended to where the new line-segment comes up against a pre-existing line-segment (drawn by either player) or up against the circumference of the circle. A player's line-segment may pass through a pre-existing line-segment. But each time a player crosses a line-segment with another line-segment, his score is halved. No line-segments may pass outside of the circle.

The second player to move in a full-move then must bounce the same direction, left or right, as the other player did, but relative to the direction this player's own last segment was traveling. And he draws his segment from where his own last line-segment ended to where his new line-segment comes up against another pre-existing segment or up against the circle's circumference. Again, his segment may pass through a pre-existing line-segment (but not pass through the circle's circumference), but doing so halves his score each time he does it.

After a predetermined number of full-moves (such as 10), each player's score = the length of that player's final line-segment divided by 2^(the number of lines crossed by that player).

Largest score wins.

Note: To be clear, there will be two "pathways" within the circle: One pathway belonging to each player, and each pathway made up of the series of connected line-segments drawn by that player.

Also, line-segments may not coincide, except at the points where they intersect.

Thanks,
Leroy Quet

Lengths Of Lengths Of Lengths Game

2010-09-18T06:02:00.001-06:00

This game is for any plural number of players.

There are a predetermined number of rounds. Each player plays the same number of rounds as the offense player.

In a round: A list is made of 0's and 1's, starting at empty-set. The players take turns each appending either a 0 or a 1 to the end (right side) of the list. The total number of digits in the round's list is n, where n is a predetermined number that is a multiple of the number of players.
(n is the same for all rounds.)

After the list of 0's or 1's is made, we determine the score.

We now form a series of lists.

(*) If the new list consists entirely of 1's, then the offense player gets the number of 1's added to his/her score. And the round is over. (If the round is over, then go to **.)

If there is at least one number not equal to 1 in the latest list: Below the last list made, write a new list consisting, in order, of the lengths of the runs of similarly-valued numbers from the previous list.

Go to (*).

(**) Change who is the offense player. Start a new round.

After all rounds have been played, the player with the largest score wins.

Sample round: n = 25:

1010110011101000111001011

1,1,1,1,2,2,3,1,1,3,3,2,1,1,2

4, 2,1,2,2,1,2,1

1,1,1,2,1,1,1

3,1,3

1,1,1

Offense gets 3 points.

Thanks,
Leroy Quet

Plusses And Minuses Grid Game

2010-09-08T13:25:00.000-06:00

This game is for two players.

Start with an n-by-n grid drawn on paper. I suggest that n be an odd integer >= 9.

The first part of the game doesn't involve the grid. In this part of the game, the players take turns each contributing to their own "prediction list" of +'s and -'a. On each move, a player appends to the end of their list either a + or a -. Each player is aware of the other player's list as it is being made. After both players' lists are n-1 symbols long, this part of the game is over. (See below for the significance of the lists.)

I suggest at this point that a dot be put at the lower left corner of the grid so as to keep this corner straight from the others.

In the next part of the game, the players take turns, each move filling in an empty square of the grid. The filled in square must not be in the same row or column as any other filled in square.

(I suggest that the player who moved first in the first part of the game moves second in the second part of the game. Just to be fair.)

After n squares total have been filled in, this part of the game is over.
(There will be exactly one filled in square in each row and in each column.)

Now, we determine the scores.
Take the grid, with the dot in the lower left corner.
Going from the left to the right, the filled in squares represent a permutation P = (p(1),p(2),...p(n)) of (1,2,3,...,n). Player 1 forms a "truth list" of +'s and -'s, where the kth symbol is the sign of p(k+1)-p(k).

Going from bottom to the top, the filled in squares represent a permutation Q = (q(1),q(2),...q(n)) of (1,2,3,...,n), where Q is the inverse permutation of P. Player 2 forms a truth list of +'s and -'s, where the kth symbol is the sign of q(k+1)-q(k).

(The lower-left square represents 1 both in respect to permutation P and permutation Q. And the upper right square represents n for both permutations.)

Write each players truth list below their prediction list so that respective signs are lined up. Each player gets a point for each corresponding pair of signs that match.

Largest score wins.

Here is a sample game:

n = 8.

Grid:
. * . . . . . .
. . . . . . . *
. . . . . . * .
. . . . . * . .
. . * . . . . .
. . . . * . . .
. . . * . . . .
* . . . . . . .

Player 1's prediction list:
+ - + - + - +
Player 1's truth list:
+ - - + + + +
The first pair, second pair, 5th pair, and 7th pair match. So, player 1 gets 4 points.

Player 2's prediction list:
+ + - + + - +
Player 2's truth list:
+ + - + + + -
The first pair, second pair, 3rd pair, 4th pair, and 5th pair match. So, player 2 gets 5 points.

Player 2 wins.

Thanks,
Leroy Quet

Add/Residue Game

2010-08-31T11:44:00.001-06:00

A game for any plural number of players:

Every player "gets" the numbers, twice each, of 1 to n -- where n is some pre-determined positive integer -- to play with; the integers 1 to n placed in each player's "mod pile", and the numbers 1 to n placed in each player's "add pile".

(A pile may simply be a list of integers that the player crosses off as each integer is played. Or if n = 13, for instance, and there are two players, then there can be 4-total actual piles where the numbers are represented by playing cards of certain suits.)

When the game begins, we have the initial value of the variable m, m(0), equal to 0.

The players take turns moving.
On a move, the player moving can choose a number from either their mod pile or their add pile. Let this number be c(k), when on the kth move of the game. Each number can only be picked once from a given pile. If playing with playing cards, place the card representing c(k) in the "used pile". Only pick from cards in your mod or add piles, not in the used pile.

On the kth move of the game, the player, depending on whether they chose their number from the add pile or the mod pile, sets m(k) to equal:

m(k) = m(k-1) + c(k), if c is chosen from the add pile.

m(k) = m(k-1) (mod c(k)), if c is chose from the mod pile.
0 <= m(k) <= c(k)-1.

If, and only if, c(k) was chosen from the mod pile, then m(k) is added to the moving player's score.

Play until the numbers from both piles of all the players are picked.
(There will be 2*n*(number of players) moves.)

Largest score wins.

Thanks,
Leroy Quet

Subsequence -- A Number Definition Game

2010-08-12T11:14:00.000-06:00

This is a game for any plural number of players.

In the first part of the game the players take turns, each player picking any one nonnegative integer to be appended to the end of a growing list of integers. The first part of the game is over when m integers are picked, where m is a positive integer agreed upon before the game by the players, and where m is a multiple of the number of players.

Let the finite sequence of integers be {a(k)}, 1 <= k <= m.

The players, in the second part of the game, take turns each coming up with a definition that describes some subsequence of {a(k)}. If the last integer in the subsequence defined by the previous player is a(j), then the currently moving player tries to find a definition that defines
(a(j+1), a(j+2), a(j+3),..., a(j+n)),
for some positive integer n. (j=0 when the first player first moves.)

The subsequence must be the first n consecutive terms of a sequence of integers defined by "the definition". The definition must be of the form "b(k) =..." (= an explicit function of k and/or of previous terms of b). The definition may only contain:any of the ten numerical digits
(
)
+
-
/ (divide, allowing fractional quotient)
\ (divide by expression following the \, then take the integer part)
* (multiply)
^ (that which follows the ^ is an exponent)
k (the index of the term), and/or
b (as in b(k-1) [= a previous term of {b(k)}], for use in recursions).

Again, a(j+k) = b(k), for all k where 1 <= k <= n.

The currently moving player gets added to their score on a move:

r*n - (the number of characters that occur in their definition after the =),

where r is some positive integer constant decided ahead of time by the players, such as r = 10.

When all m integers have been described, then the game is over.

Largest score wins.

(It should be noted that r should be small enough such that a player creating, say, a polynomial P(k) which outputs a(j+k), for all k where 1<=k <= m-j, would end up only losing points. If the first player to move creates such a polynomial for all k, 1<= k <= m, then they should receive a negative score, and all other players tie for first place each with a score of 0.)

Any problems you can see with the game? (Of course there are.)

Thanks,
Leroy Quet

DominatriX -- Simple grid game

2010-07-29T11:03:00.000-06:00

Game. Any plural number of players. n-by-n grid. (I suggest for a 2-player game an n of about 4 to 6 for beginners.)

The first player to move puts an x in any square.

The players take turns. (The player who last drew an x is player A. The player who is now choosing where to put an x is player B.)
The players take turns who is player A and who is player B.
After player A draws an x, she then tells player B how many squares from player A's recent x that player B can put his x. (This distance is k squares.)
Player B then chooses a direction (up, down, right, or left-- whatever is possible, given the edges of the grid) from player A's x that he shall put the new x.
Player B can only put an x in an empty square. And this x must be the number (k) of squares dictated by player A from player A's last x (as I already said).
And, if possible, player A MUST dictate the distance (k) to an empty square from her x.

Last player able to move is the winner.

So, to be the last player, you want to move into an empty square that is the last empty square in that column and row.

I actually played this game with someone else. (I admit, I hardly ever play my games with other people before publishing them. I have a hard time finding people willing to play.)

For novices, I found, the game goes along without too much use of strategy until the last few moves.
I would guess that if the players are more advanced, then strategy would come in sooner. But maybe not. Maybe, maybe, it doesn't really matter how good you are until the last few moves.

Thanks,
Leroy Quet

Subdivide And Acquire Game

2010-07-26T11:34:00.001-06:00

Here is a game for any plural number of players. Start with an n-by-n grid drawn on paper, where n is larger if there are more players.
Each player has a colored pencil of unique color.

In the first part of the game, players take turns drawing straight line-segments -- one line-segment each turn. A player can draw a line-segment from either the edge of the grid at a vertex, or from the end of another line-segment. (Multiple line-segments can join at one point.) The line-segments are drawn to any vertex of the grid (either empty or already occupied by a line segment), such that no line-segment is drawn through another segment or along another segment or through a vertex occupied by line-segments. (Although, as I said, a line-segment may end at a vertex already occupied by another segment.)

After the grid is subdivided into n*(number of players) sections, the second part of the game begins.
Players take turns filling in sections of the grid with their colored pencils. After each player fills in n sections, the score is determined.

Players add up the total area of all the sections in each player's color, with the area of a grid-square being 1. (This may be tricky because some sections will most probably have non-integer areas.)

Let a player's total area be m; then the winner is the player where
number of divisors of floor(m) is the SMALLEST.

Thanks,
Leroy Quet

Transform -- The Game

2010-07-23T04:37:00.000-06:00

This is a game for any plural number of players.
(No grids this time, sorry.)

There are a number of rounds in this game. The number of rounds is a multiple of the number of players. The players take turns being "the permutator", where each player is the permutator the same number of rounds.

Before starting any of the rounds, the players agree on a positive integer n. (n is the same value for all rounds in the game.)

On a round, all of the players (including the permutator) start the round by each coming up with an ordered list of n integers (positive, negative, or 0). Each player's numbers must be distinct, in that no integer occurs more than once in a particular player's list. The players each keep their lists secret from the other players for now.

Let player p's list for any particular round be {b(p,k)}, k = 1,2,3,...n.

Next, all players who are not the permutator take turns choosing terms of a list of n distinct integers(positive, negative, or zero; no integer more than once). (So, if the number of players is m, I guess to be fair, n should be a multiple of (m-1).)

Let this list be {a'(k)}, k = 1,2,3,..n.

Then, after the list is complete, the permutator forms any permutation {a(k)} of {a'(k)}.

Then everyone reveals their b-lists.

Each player p forms a sequence of n integers in this manner:

c(p,k) = sum{j=1 to n} a(n+1-j) b(p,j). k = 1,2,3...,n.

Player p's score for this round is the number of primes in {c(p,k)}.

After all the rounds are played, the players add up their scores for all the rounds. The player with the largest grand score wins.

In a variation, instead of the number of primes being the criterion for scoring, the players decide amongst themselves before each round what will be the criterion for a number in the c-list to score. Fibonacci numbers? Squares? Where each c(p,k) is coprime to c(p,k-1)?
Or maybe the choice of criterion should be totally up to the permutator (with veto power from the other players), and expressed at the beginning of each round before the b-lists are constructed.

Thanks,
Leroy Quet

TraverX

2010-07-16T09:54:00.000-06:00

This is a game for any plural number of players. Start with a grid of n-by-n squares ((n+1)-by-(n+1) lines) drawn on paper, where n is even, and where I suggest that n is >= 12.

At the beginning of the game, a small x is drawn at the intersection of the middle horizontal line and middle vertical line.
Players take turns moving. On the k-th move (the kth move considering all the players' moves together) the player "traverses" j(k) = (k-1)(mod(n-1))+2 intersections from where the last player last put an x.
(So, for k = 1,2,3,4,...,n-1,n,n+1,n+2..., the number of intersections traversed is 2,3,4,5,...,n,2,3,4,..., repeating 2 through n.)
(On the first move, the first player traverses 2 positions from the central x.)

The player can "traverse" j(k) intersections in the direction of either right, left, up, or down, and then may change direction at any time at most once, and traverse perpendicularly to their initial direction for the remainder of the j(k) intersections traversed. The player then places an x at the intersection they land upon. It is only acceptable for players to land upon (at the j(k)th intersection traversed) an intersection without an x already drawn upon it. Players may, though, traverse over intersections with x's already on them, or not.

After a player writes down an x, he/she gets a point (points are bad in this game) for every other x already written on the same vertical line and same horizontal line as their x.

The game continues until any player cannot move anywhere (given j(k) and the lack of available intersections).

The player with the SMALLEST score wins.

Thanks,
Leroy Quet

Vertical/Horizontal Guessing Game

2010-07-05T13:09:00.000-06:00

Here is a game for two players, and, you guessed it, it uses an n-by-n grid drawn on paper. (I suggest an n of at least 8.)

First, the players each secretly guess how many squares will be filled in before the game terminates. Each player writes their guess down and hides the guess from their opponent.

Next, fill in with a pen/pencil the center square (if n is odd) or one of the 4 center squares (if n is even).

(*)The players then both secretly choose a direction. Player 1 chooses either up, vertically-steady, or down. Player 2 chooses either left, horizontally-steady, or right. The players each write down their choices.

Then the players both reveal their choices simultaneously.

The next square filled in has the vertical direction chosen by player 1 and the horizontal direction chosen by player 2 from the last square filled in; and this next square filled in is adjacent (touching on a side or on a corner) to the last square filled in. (There are 9 possible combinations of directions, including not changing the square at all {when both players pick steady}.)
Note: If the last filled in square is on the edge of the grid, then this limits what directions can be stated by one or both of the players.

If that adjacent square is already filled in, then the game is over.

But if that square is empty before being filled in on the current move, then the game continues. (Go to *.)

When the game is over, the winner is the player whose guess for the number of squares filled in is closest to (either greater than or lesser than or equal to) the actual number of squares filled in.

(By the way, if both players choose the direction 'steady' at the same time, the game ends then, of course.)

Thanks,
Leroy Quet

Predicate -- A Game of Numbers And Creativity

2010-06-23T12:44:00.002-06:00

This is a simple game that has the potential to go horribly wrong...

This game is for any plural number of players. Let the number of players be m.

Each player has a different colored pen/pencil/crayon.

Make a number line with the positions immediately beneath it labeled in order with 1 through m*n, where n is some positive integer decided ahead of time by the players.

The players take turns. On a PLAYER'S k_th move, he/she writes (with the pen/pencil/crayon of his own color) the number k just above any one of the empty positions along the number line.
After every player has written n -- after a total of m*n moves, and the number line is filled up -- the next part of the game begins.
(When the first part of the game is complete, every integer k occurs exactly m times on the top of the number line.)

But before writing down the numbers, each player comes up with a rule for scoring points. The players all write down their rules, and only reveal them after the number line has been filled with numbers.

Each rule completes this sentence:

A point is scored for a player for every integer in the player's color where _______.

The rule must be based on the position number (below the line) of the integer (above the line) being tested , and/or on the neighboring integers written during play (above the line).

A rule must NOT be based on the colors of the integers or on any external variables.

The rules may use any mathematics the players personally choose.

All the rules apply to all the players' numbers fairly.
In other words, the players EACH come up with a rule, and all the rules are used to test all the players' numbers, and the points obtained (in respect to all the rules) by each player are summed.

An example of some rules:

A point is scored for a player for every integer k in the player's color where _______.
* k is next to exactly one integer of opposite parity.
* k = the number of divisors of its position-number.
* k divides the sum of its immediate neighbors.
* k is coprime to the sum of all numbers to its left.

(My examples use basic number theory, but you can involve other branches of mathematics.)

Largest score wins.

Variation:
Play on a grid instead of number line.
Involve the number of the column and/or the number of the row of each number being tested, as well as neighboring numbers, possibly.

Any unforeseen (by me) problems with this game?

Update: (6/30/10) I changed this game so that the rules the players choose are written down BEFORE the numbers are placed along the number line. The rules are kept secret until after the number line is filled with numbers.

Thanks,
Leroy Quet

Upward-Rightward Game

2010-06-10T09:37:00.001-06:00

Here is a game for 2 players, using an n-by-n grid drawn on paper.

Each player has a pen/pencil of a color different than their opponent's color.

First, the players take turns filling in the squares, one square per move in this part of the game.
The first player fills in the lower left square with his/her color. Thereafter, each player fills in the square (with the player's own color) either immediately to the right of or immediately above the last square filled in by the previous player.

This crooked "line" of squares continues until it reaches the upper right square. (So the last few squares filled in this way may be forced to be in the top row or most rightward column.)

Then, in the second part of the game, the players take turns filling in a number of squares. On a single move, a player fills in any empty square (of any color) above or right of any square filled in before in the game. Then, in that same move, the player may fill in any number of empty squares where each square filled in that move is above or right of the square filled in previously by the same player during that move. This crooked line of squares may terminate at any time, but must contain at least one square.

When all n^2 squares of the grid are filled in, the game is over.

Player 1 gets as a score:
sum{k=1 to n} (product of lengths of runs of squares in row k)

Player 2 gets as a score:
sum{k=1 to n} (product of lengths of runs of squares in column k)

A "run" contains consecutive squares (in a specific row or column) all of the same color (either color), bounded on each side by squares of the opposite color (or bounded by the end of the row/column).

Largest score wins.

Here is a completed sample 6-by-6 game:
x x x o X O
x x x o O x
o o o O X o
o X O X o o
o O o o o x
O X o o o o

(Capital X and O are drawn during first part of game. Lower-case letters are drawn during second part of game. Sample game played without strategy.)

Player 1 gets (sum over rows):

3*1*1*1 + 3*2*1 + 4*1*1 + 1*1*1*1*2 + 5*1 + 1*1*4 = 24 points.

Player 2 gets (sum over columns):

2*4 + 2*1*1*1*1 + 2*4 + 3*1*2 + 1*1*1*3 + 1*1*2*1*1 = 29 points.

Player 2 wins.

And of course, players can play with one pen, and "fill" the squares each with a different symbol.

Thanks,
Leroy Quet

Deja Vu Divisors

2010-06-09T10:13:00.000-06:00

Here is a game for two players.

The players take turns picking any integer from 1,2,3,...., r that has not been picked (by either player) previously in the game, where r is some large integer (such as 1000 or 10000 or more).

Let this picked number be m. (m was picked by the player temporarily called the "provider".)
The same player then picks any integer k where 1 <= k <= m. (k may have been picked earlier any number of times in the game.)

The other player (the "finder") then tries to find any positive integer n not equal to m (n can be arbitrarily large and either picked previously during the game or not) such that:

Both d(m) = d(n) and d(m+k) = d(n+k), where d(j) is the number of divisors of j.

The finder may also provide a proof that no such n (not equal to m) exists.

If the finder either finds an n or proves there is no such n fitting the conditions, then the finder gets a point.

Players then switch who is the provider and who is the finder.

Players play an even predetermined number of moves, and the player with the largest score wins.

Thanks,
Leroy Quet

Bidirectional Line Game

2010-05-23T13:19:00.000-06:00

Here is a game for two players.
Needed, an n-by n grid drawn lightly on a piece of paper. Two pens or pencils of different colors.
(I suggest n be at least 6.)

In this game, line segments are each drawn from some vertex of the grid (along the lightly drawn grid line) to an adjacent (immediately above, right of, below, left of) vertex.

On the first move, the first player draws a line segment with one of the pens from any vertex to any adjacent vertex.
Then the other player draws a line segment with the other pen from where the first line segment started to any adjacent vertex, as long as the two line segments aren't drawn to (end at) the same vertex.

Thereafter, both players on each of their moves may draw a line segment of EITHER color from the vertex last drawn to by a line segment of that color. The line segment is drawn to any vertex such that no line-segments coincide (except perhaps at a single point). So, players may choose which end of the growing 'line' (a collection of line segments placed end to end) to extend with a line segment on each of their moves, provided that the color of any new segment matches the segment it is attached to.

The line may pass through itself. (Example: Two older consecutive vertical segments are conjoined at a vertex. Then later on, a newer line segment comes from the left and meets at that vertex, then the lastly drawn of these four segments proceeds from the vertex rightward.) (The older line segments may or may not be the same color as the newer segments.)

The line may also "bounce" off of a corner, making a new corner.(Example: An older segment proceeds upwardly to a vertex, then the next older segment proceeds to the right. Then later on, a newer segment comes from the left to meet at that vertex, and finally the last segment is drawn upwardly from that vertex.) (Again, the older line segments may or may not be the same color as the newer line segments.)

Continue the game until one of the ends of the total line cannot be drawn anymore. (This will be either at the edge of the grid or perhaps if, not necessarily when, the two ends of the line meet.)

The winner of the game has the most line segments of the color he drew his first segment in.

Thanks,
Leroy Quet

Scrambled Number-Row Game

2010-05-08T10:02:00.000-06:00

Here is a game for any number of players. (I wonder, though, if having an even number of players gives an advantage to some player(s).)

Needed: A deck of n flash cards labeled 1, 2, 3, ..., n, one number per card, where n is a multiple of the number of players.

The cards are placed face-up in order in a row between the players.

Players take turns, switching one pair of cards each move.
On the m-th move of the game, a player switches the positions the card labeled with the number m and any other card.

The player moving gets a point if both:

One of these cards he/she switched (either card; let k be the number on this card) is adjacent to a card with a number coprime to k, if the card is now at the end of the row of cards, or card k is now between two cards both with numbers coprime to k;
(In other words, card k is NOT non-coprime to any card it is now adjacent to.)

AND

The other card switched (with the number j on it) is non-coprime to exactly one number it is now adjacent to. (Either the other number that card-j is adjacent to is coprime to j, if card j is not at the end of the row, or card j is at the end of the row.)

After n total moves, the player with the most points wins.

Clarification: m equals either k or j, not both.
Each move results in a permutation of the numbers 1 through n.

I suggest that n be large enough to make this game relatively interesting, of course.

Thanks,
Leroy Quet

Circle Intersection Game

2010-04-20T09:07:00.001-06:00

Here is a game for any plural number of players.

Needed: Blank piece of paper, compass.

First, with the compass, draw two circles of the same radius, with each circle passing through the other's center.

Players take turns. On a player's turn, he/she draws a circle (with the compass, of course) of any positive radius such that the center of the circle is at the intersection of at least two previously-drawn circles. The number of intersections passed through by this new circle is added to the player's score -- where each intersection is of at least two previously drawn circles and of the circle just drawn by the player, for a total of AT LEAST THREE circles per intersection (including the circle just drawn by the player). Note: The number of intersections (each of any number of circles), not the total number of circles in these intersections, is added to the player's score.

No two circles may have both the same radius and same center. (No two circles can coincide completely.)

The first player to reach a predetermined number of points wins.
(I suggest a higher goal score for a larger piece of paper and larger initial circles. I suggest a lower goal score if there a more than just a couple players.)

Note: If 3 or more circles ALMOST intersect at a point -- but their exact point of coinciding is uncertain because the circles were poorly drawn -- use geometric theorems to determine if indeed the three or more circles coincide *officially* at that point.

Thanks,
Leroy Quet

Lines-From-Lines Game

2010-04-11T09:56:00.000-06:00

Sorry if this is unoriginal.

Here is a game for two players. It is played on an n-by-n grid drawn (practically perfectly) on paper. I suggest that n be at least 8.

Note: (The lines of the grid do not come into play here. Only the "grid-vertices" {where the lines of THE GRID intersect} are important as far as the grid is concerned.)
By "lines" below, I am referring to straight line-segments drawn by players during play. I suggest these lines be drawn with a straight-edge.

The first player to move draws a line from any grid-vertex to any other grid-vertex.

The players thereafter take turns drawing lines, one line per move, such that:
* each line starts at a grid-vertex intersected by any previously drawn line (drawn by either player).
* each line ends at any grid-vertex not yet touched by a line.
* no line crosses another line or coincides with another line.
* no line starts/ends at or crosses another line's end/start-point (whether or not yet another line passes completely through that vertex).

If you can move, you must. (The other player can help you find allowable moves.)

The LAST player to be able to move LOSES.

And let me clarify things, in case I am a bad writer.
A line may PASS THROUGH a vertex at most once.
A line may START OR END at a vertex at most once.

The same vertex may have one line passing through it and another
single line terminating (ending or starting) there.

There must be an easy strategy to always win. Anyone know of one?

Thanks,
Leroy Quet

Simple Number-Picking Games

2010-03-26T13:51:00.001-06:00

This is really a post about two games, one simple, and the other simpler.

The simple game first.

For any plural number of players.

Players take turns. On each turn, a player picks one number from (2,3,4,...,n) that has not yet been picked in the game. (n is, say, 100.)

Each number, after the first pick, must NOT be coprime to the number picked in the previous move by the last player to move.
Or
If the last number picked in the game was a prime AND no multiples of that prime exist among the numbers that have not yet been picked, then the player may pick any prime from the primes that have not yet been picked.

The last player able to move wins.

--

Simpler game:

For 2 players.

As before, players take turns. On each turn, a player picks one number from (2,3,4,...,n) that has not yet been picked in the game.
Each number, after the first pick, must NOT be coprime to the number picked in the previous move by the last player to move.

If a player can move, the player must move.

The last player to move LOSES.

Thanks,
Leroy Quet

Zigzag Edge Game

2010-03-17T15:01:00.000-06:00

A game for two players:

Start with a grid of (n-1)-by-(n-1) squares, or n-by-n lines. I suggest that n be at least 12.

The players take turns placing dots on the grid, one dot each move at any intersection of the grid that does not yet have a dot on either of the two intersecting (horizontal and vertical) grid-lines.

After n dots are placed -- and the dots represent a permutation of (1,2,3,...n) -- the strategy portion of the game is over.

Either of the players then draws (n-1) straight line-segments (called "permutation lines"), starting at the leftmost dot, a line-segment drawn between each dot and the dot on the vertical grid-line immediately to the right. (So, you get a zigzag, in lots of cases.)

Then, either player draws (n-1) straight line-segments (also called permutation lines), starting at the topmost dot, a line-segment drawn between each dot and the dot on the horizontal grid-line immediately below.

Now, the two intersecting zigzags are inclosed within a polygon (not talking about the convex hull), the perimeter of which is made up of permutation lines and/or parts of permutation lines.
Call each straight line-segment along this perimeter an "edge line". Two permutation lines that meet at one vertex AND have the same slope are considered to both be part of one edge-line. Two edge lines may come together somewhere other than at a dot (in which case an edge line takes up only part of a permutation line).

Player 1 gets as a score the number of edge lines from the leftmost dot to the rightmost dot along the TOP of the bounding polygon's perimeter.

Player 2 gets as a score the number of edge lines from the leftmost dot to the rightmost dot along the BOTTOM of the bounding polygon's perimeter.

Largest score wins.

Note: The bounding polygon may narrow to a single point. Four edge-lines are considered to meet at such a point, even though only two permutation lines cross there.

What is a strategy for this game if you are player 1, or if you are player 2?

Example game:
Let us say we have a grid of 9-by-9 lines.
Label the grid's vertical lines 1 to 9, starting at the bottom. And label the grid's horizontal lines 1 through 9, starting on the left.
During play, the dots are drawn at these grid-line intersections:
(1,6), (2,7), (3,8), (4,4), (5,5), (6,9), (7,1), (8,3), (9,2).

Player 1 gets 5 points. Player 2 gets 5 points. A tie.

Note that the permutation-line segments connecting (1,6) to (2,7) to (3,8) count as one edge line. Also note that the permutation line (6,9) to (7,1) and the permutation line (4,4) to (8,3) intersect within a grid-square, and that we have 4 edge-lines (2 edge lines to player 1's score, 2 edge lines to player 2's score) meeting at this intersection.

Thanks,
Leroy Quet

Arcs And Marks

2010-02-25T12:08:00.001-07:00

Needed: blank paper, pencil and compass (the circle-drawing kind), maybe a protractor.

First, draw a circle on a piece of paper, relatively large.

Let m be a multiple of the number of players playing this game. m should be >= 8, at least, I suggest.

Draw m pencil-marks EVENLY SPACED along the circumference of the circle. (This is why you might need a protractor, if you can't just use the compass and straightedge to accomplish this by geometric construction.)

There will be m moves in the game (or in the round).

Players alternately take turns drawing arcs, drawing one arc with the compass on each move. Each arc must be drawn within the circle's interior from the circle's edge back to the circle's edge. On a move, a player uses the appropriate mark as the center of the circular arc that passes through two other marks. The two marks (which the arc intersects the main circle at) must be equidistant from the arc's center mark, and that distant must be nonzero.
The first player starts at any mark to make the center of his arc; and on each move after the first, the moving player uses for her arc's center the mark immediately clockwise from the mark used for the previous player's arc-center.

Count the number of preexisting arcs (not including the "arc" of the main circle) that the moving player's arc intersects or touches. Each player has a running-total of the number of arcs his/her arcs passed through.

The winner has the FEWEST total number of earlier-drawn arcs (drawn by any player) intersected or touched by their arcs.

Notes: An arc may touch/intersect other single arcs more than once each, but each such incident only counts once towards the number of arcs intersected or touched.

Also, multiple arcs touching at a point are counted as touching all other arcs coming together at that point, no matter which arcs are "behind" or "ahead of" which other arcs.

There seems to be an advantage for who moves first, and a disadvantage for who moves last. So, maybe multiple rounds should be played, a different round where each player is the first player to move. (Draw a new circle with the same number of marks each round.) Then the total scores for the rounds are added up for the grand score for each player. Players try to minimize their grand scores, of course.

Thanks,
Leroy Quet

An Integer Sequence Game

2010-02-16T09:50:00.000-07:00

This is a game for any number of players.

Needed: Pencil/pen, paper, calculator (with long display) perhaps. (Maybe this game could be played via a computer running the appropriate program.)

Start by writing down the integers 1, 2, 3,..., n, where n is at least 8 or more if the number of players is 2, I suggest. n is larger if there are more than 2 players.
This list of integers is called the "r-list".

The variable m starts the game with the value 1. In other words, m(0) = 1.

Players take turns. On the kth move (the kth move among all players together), the moving player lets r(k) = any uncircled integer from the r-list.
The player then circles that number.

m(k) is the value of m after the kth move.
Let m(k) =
r(k)*m(k-1) + (number of composites among m(0),m(1),m(2),...,m(k-1)).

Add to the moving player's score the largest value from m(0),m(1),m(2),...m(k-1) that divides m(k).

The move is complete when the moving player writes down m(k) at the end of the growing list of the values of m.

Players keep taking turns until k = n.

---

Example game, n = 8: (I may have made a mistake with my math.)
m(0) = 1
r(1) = 2; m(1) = 2*1+0 = 2. (Prime.)
Moving player gets 1 added to score.
r(2) = 8; m(2) = 8*2+0 = 16. (Composite.)
Moving player gets 2 added to score.
r(3) = 3; m(3) = 16*3+1 = 49. (Composite.)
Moving player gets 1 added to score.
r(4) = 5; m(4) = 49*5+2 = 247. (Composite.)
Moving player gets 1 added to score.
r(5) = 1; m(5) = 247*1+3 = 250. (Composite.)
Moving player gets 2 added to score.
r(6) = 4; m(6) = 250*4+4 = 1004. (Composite.)
Moving player gets 2 added to score.
r(7) = 6; m(7) = 1004*6+5 = 6029. (Prime)
Moving player gets 1 added to score.
r(8) = 7; m(8) = 6029*7+5 = 42208. (Composite, but this does not matter.)
Moving player gets 16 added to score.

---

How does the sequence {a(k)} begin, letting a(n) = the largest possible score for a 1-person game where the r-list contains the first n positive integers?

Thanks,
Leroy Quet

Labyrinthine Loop

2010-02-08T10:20:00.002-07:00

Here is an (unoriginal) game for any plural number of players.

The game consists of rounds, where every player is the "offense-player" the same predetermined number of rounds.

At the beginning of each round, draw an array of dots (vertices of a grid) on a piece of paper, n rows of dots by n columns. (n is a predetermined integer, which is the same for all rounds. I suggest n be >= 6.)
There is a new array of dots for each round.

At the beginning of the round, the offense-player moves first, connecting any pair of adjacent dots with a straight line-segment. (By "adjacent", one dot must be one of the closest dots to the other dot, and in the direction of E, SE, S, SW, W, NW, N, or NE.)

Players thereafter continue to take turns. On a turn, a player connects (with a straight line-segment) any dot that has AT LEAST ONE line-segment connected to it already, to any ADJACENT dot that has NO line-segments connected to it.
Again: By "adjacent dots", it is meant that one dot is one of the closest to a second dot, where the two dots are in the direction of either E, SE, S, SW, W, NW, N, or NE to each other.

Diagonal line-segments MAY cross each other.

Players continue taking turns until all dots have line-segments connecting to them.
(ie. Players continue taking turns until a total of n^2 -1 line segments are drawn in a round.)

Then, lastly in the round, the offense-player connects any pair of unconnected adjacent dots with a line-segment.

The offense-player's score for that round is the number of dots in the (single) closed loop of line-segments (including the line-segment the offense-player just drew).
(The "loop" is the simple path from dot to dot that connects back to its starting point. The loop does not including dead-ends, of course.)

After each player has played offense the same predetermined number of rounds, then each player adds up his/her scores from those rounds she/he was offense to get her/his grand total. The player with the highest grand total wins.

PS: It should be noted that when determining the shape of the loop created in the final step of a round, each "line-segment" in the final loop goes strictly from a dot to another dot. When two diagonal line-segments cross, they are considered, for our purposes, to not be touching -- one segment goes "over" another. As a result, there is always one and only one loop made each round.

Thanks,
Leroy Quet

Numbers To Number

2010-02-05T08:25:00.000-07:00

This is a game for 2 players. Needed: 3 blank pieces of paper, one for each player and one common piece of paper. 2 pens/pencils, one for each player.

Let m be an integer decided on by both players. m should probably be >= 6. Players also decide who is player 1 and who is player 2 at the beginning of the game.

At the beginning of each round, that round's "binary list" is blank. (The binary list is a series of 1's, 0's and _'s written on the common piece of paper, one new binary list for each round.)

On a turn, both players secretly write down on their piece of paper any number from 1 to 2^m -1 that has yet to be written down by that player in the game (in any previous round or in the current round).

The two players' numbers are then revealed to each other.

If player 1's number is > player 2's number, then append a 1 to the right side of that round's binary list.

If player 1's number is < player 2's number, then append a 0 to the right side of that round's binary list.

If player 1's number is = player 2's number, then append a _ (underscore) to the right side of that round's binary list.

After m moves (where one move is both players moving simultaneously) have passed in the round, the round is over.
Treat the binary list as a set of binary numbers, with each _ treated as either a 0 or a 1. Convert each binary integer to decimal.
So, the binary list represents 2^(number of _'s) different integers.
Let the set of decimal integers for that round be D.

So, for instance, if the binary list (m=8) looks like this:

01_001_1,

then D for that round contains:
69, 71, 101, and 103.

Each player gets a point for every element of D that was an integer played by the player IN THAT ROUND.

After a round is over, the player crosses off all the numbers played by that player in the round (so that the player can tell the difference between numbers played in that round and numbers player in earlier rounds).

A game consists of floor((2^m -1) /m) rounds.

Add up scores from all rounds to get each player's grand score.
Highest grand score wins.

Thanks,
Leroy Quet

Move In Synch, In Opposition

2009-12-22T14:30:00.003-07:00

Here is a game for 2 players.

Needed: An n-by-n grid drawn on paper (with an n of at least 8, I suggest).
Two markers (such as coins), one for each player. The markers should be small enough to both fit in one square of the grid (possibly by stacking them).
Two pencils/pens of different colors.

The players start the game by each placing their marker in any square of the grid different from where their opponent placed her/his marker.
The players then mark the squares they start in with an X (or with different symbols, especially if the colors of their pens/pencils are similar).

A "move" consists of both players moving one square each.
Players alternate, taking turns who is the "directioner" and who is the "decider".

On a move, first the decider decides if the players will move in opposition or in synch, and announces this decision. Then the directioner decides which of the 8 directions (up, down, left, right, or diagonally) the directioner will then move, then he/she moves his/her marker.

If the decider decided that the players move in synch, then the decider must move his/her marker one square in the same direction (from his/her current position) that the directioner moved her/his piece (from the directioner's previous position).

If the decider decided that the players move in opposition, then the decider must move his/her marker one square in the exact opposite direction (from his/her current position) that the directioner moved her/his piece (from the directioner's previous position).

Based on where the players are at the beginning of a move, the directioner must pick a direction where both players stay on the board.

If any player moves his/her marker onto an empty square, then that player draws his/her symbol in the empty square. But if both players move onto the same empty square, or a player moves onto a square with a symbol already in it, then no symbol is drawn by that player on that move.

When all squares are filled up with symbols, or when one player has more symbols than the number of symbols his/her opponent has + the number of empty squares, then the game ends. The player with the most squares with his/her symbol is the winner.

A variation: (Is this more or less fun than the first version?)
The directioner picks the direction first, then the decider decides if she/he will move his/her piece in synch or in opposition. (The directioner, in this version, only moves based on whether he/she can stay on the board, then the decider must move to stay on the board.)

Note: In either variation, if for some reason the decider can't move on a move, the he/she just stays put that move.

Thanks,
Leroy Quet

Simple Card Game Of Ups And Downs

2009-12-14T07:30:00.000-07:00

This is a card game for two players.

Start with a deck of 2n cards labeled 1 through 2n, one distinct number per card. (n is some number, such as 10.)

Shuffle the deck, and deal n cards to each player.

Each player secretly arranges their n cards in any order they choose.

The players then, without further arranging their cards, take turns alternately placing cards, in order, face-up onto a common pile between the players.

Important: When the players are placing their cards in the central pile, they MUST place them in the order they originally ordered them, from left to right in their hand (or from top to bottom in their own face-down card pile).

If a card placed in the central pile is greater in numerical value than the last card placed down by the previous player, then player 1 gets a point.

If a card placed in the central pile is smaller in numerical value than the last card placed down by the previous player, then player 2 gets a point.

Continue until all 2n cards have been placed in the central pile.

The player with the greatest number of points wins.

-

Variation: Play with a standard deck of 52 cards. (No jokers.) (Therefore, in this case, n = 26. Ace =1, Jack = 11, Queen = 12, King = 13.)

Same rules as before, but if a card is of the same value as the previous card, then player 1 gets a point if the newly placed card is of a red suit (hearts or diamonds), and player 2 gets a point if the newly placed card is of a black suit (ace or clubs).

Is this game unoriginal? It sounds familiar.

Thanks,
Leroy Quet

Precognition -- Card Game

2009-11-19T12:44:00.001-07:00

This is a card game for a plural number of players.

First, the cards from a standard deck (no jokers) are dealt face-down to the players, so that each player has the same number of cards.

The players then each examine the hand they have been dealt, keeping their cards secret from the other players. (Of course, if there are two players, you know the cards your opponent has are exactly those cards you don't have.) :)

All that matters in this game as far as the cards are concerned are their numerical values. (Ace = 1, Jack = 11, Queen = 12, King = 13.)

The players each secretly on their own piece of paper write down a series of letters ("U" for up, "D" for down, "S" for stay), corresponding to a predicted outcome. (See below.) The players can write down any number of these letters they each choose -- 1 letter, up to a string of letters of length equal to the number of cards.

Next, the players take turns placing cards face-up, one card per move, making a single row of cards on the table between the players. The cards are placed in the row from the left to the right. (I suggest that each card be placed on top of the card below it, being placed a little to the right so that the value of each card is showing.)

After all cards are placed in the row, the players reveal their lists of letters.

Consider the "changes" between consecutive numbers in the row of cards. Either a number goes up (U) from the previous number in the row, goes down (D), or stays the same (S). Form a list of these changes written in order from left to right.
The winner is the player with the longest string of letters that corresponds to any subset of consecutive changes within the row of cards.

For example, if we have the (short) row of cards:

2,6,5,4,7,9,7,1,2,2,5,8

And a player has "UUDDUSU",
then this corresponds to:
2,6,5,(4,7,9,7,1,2,2,5),8

because 4 to 7 is U (up), 7 to 9 is U, 9 to 7 is D (down), 7 to 1 is D, 1 to 2 is U, 2 to 2 is S (stay), and 2 to 5 is U.

If this is the longest matching string (7 letters) of U's, D's and S's by any player, then this player wins.

(Note: A player can almost always get a match, for example, by having a string of one letter U or D. But then there is a good chance someone else will have a longer matching string.)

If there are a number of players that all tied for first place, then these players play again amongst themselves as many games as necessary, eliminating players each round, so as to determine a final champion.

Thanks,
Leroy Quet

Procession

2009-11-03T14:00:00.000-07:00

This is a game for any plural number of players.
Needed: piece of paper and a pen/pencil.

Start by writing a row of n 0's on the piece of paper. (n is a positive integer decided beforehand by the players. I suggest an n between 5 and 10 for a 2 person game. Slightly more for more players.)

After writing the row of n 0's, write the value of n to the right of this row.

Next, the players take turns. On a player's move, he/she copies the row (which will be of 0's and 1's) immediately above, but with either one 1 changed to a 0, or one 0 changed to a 1. (The player can change any one digit she/he chooses, under restrictions -- see below.)

Next, that same player writes down (to the right of the row) the lengths of the runs of both 0's and 1's in the row he just wrote down.
Each "run" is made up completely of 0's or completely of 1's, and is bounded by runs of the other digit or by the edge of the row. (No two consecutive runs are of the same digit.)
It doesn't matter if a run is of 0's or 1's. All that matters in this game is where each boundary is between each run of 0's and the adjacent run of 1's.

* A player, though, cannot change a digit on his move such that the multiset of run-lengths (of the row of 0's and 1's just created) has already occurred in the game.
(A "multiset" is a list of numbers where the order of the numbers in the list is unimportant, but the number of occurrences of each number is indeed important. For example, {1,2,1,3) and (2,1,1,3) would be considered to be the same multiset, but (1,2,1,3) and (1,2,3,3) would not be the same.)

The last player able to move is the winner.

Sample game. Simple example:
(n=5)

00000 5
00010 3,1,1
10010 1,2,1,1
10011 1,2,2
(Can't do 10111 here, for example, because the run-length multiset 3,1,1 already occurred.)
00011 3,2
00001 4,1

The player who wrote 00001 wins, because 10001 (run-lengths 1,3,1), 01001 (1,1,2,1), 00101 (2,1,1,1), 00011 (3,2), and 00000 (5) each have a multiset of run-lengths that already occurred.

FYI: The total number of moves in a game is no more than the number of (unrestricted) partitions of n. (So, there is a maximum of 7 moves in an n=5 game.)

Thanks,
Leroy Quet

Dot Labeling And Connecting Game

2009-09-15T14:39:00.000-06:00

This is a game for any plural number of players.

First, draw an array of n-by-n dots, which are the vertices of an (n-1)-by-(n-1) square grid. I suggest an n of at least 9.

Each player corresponds to a color or symbol (such as a number or letter or something else).

Players take turns. On each move a player connects two different dots (not necessarily in the same row or column) with a straight line-segment, then labels ANY unlabeled dot with ANY player's color/symbol. The two dots connected by the line-segment must not have been connected together in a previous move, although they may have been connected to other dots earlier in the game.

The line-segment must terminate at the two dots, and must not pass through any intermediately positioned dots or through any other line-segment.

As soon as all dots are labeled (after n^2 moves), the game is over.

Each player gets a point for each "group" of dots with the player's color/symbol, where each group contains at least two dots connected by a line-segment. A player only gets one point for each group of dots where every dot of that group -- a group of all dots of the same symbol/color -- is accessible by traveling along the line-segments from dot to dot of the same color/symbol.

If two dots of a player's color/symbol(#) can only be connected by traveling through a dot of another player's symbol/color, then the two dots of color/symbol # are considered to be in separate groups.

(There is graph-theory terminology for what I am trying to express, but I don't feel like looking up what that is.)

(And remember, each group that earns a point for a player must contain at least two dots of the player's symbol/color.)

The player with the most points wins.

Thanks,
Leroy Quet

Grid Game: 1 to 1 to 1

2009-08-19T12:46:00.002-06:00

This is a game for, preferably, 3 or 4 players.

Start with an m-by-m grid drawn on paper. I suggest that m be at least 3 times the number of players.

Each player take turns writing odd numbers in the squares, each number being small enough that other numbers may also be written in any square if necessary.

In each player's first move, he/she writes a 1 in any empty square of the grid.

Thereafter, a player places a 3 then a 5 then a 7, etc, each number in a square. The number (2k+1) must be in the square adjacent and horizontal, vertical, or diagonal to the square (but not in the same square) where the SAME player last wrote the number (2k-1). Each number must either be written in an empty square, or be written in a square such that the new number is coprime (relatively prime) to all numbers previously written in that square (by any player).

The first player whose path of numbers visits all of her/his opponents' 1's and then lastly returns to her/his own 1 is the winner.

If a player cannot move, then he/she is out of the game.

A player may win if all other players forfeit by not being able to move.

Thanks,
Leroy Quet

PS: I have changed the rules slightly to have all the numbers be odd. -- 8-20-09

Rectangles Of Distinction

2009-08-06T14:59:00.001-06:00

This is a game for any plural number of players.

First, draw a (2n)-by-(2n) array of dots (where the dots correspond to the vertices of a grid of (2n-1)-by-(2n-1) squares), where 2n is at least 6, I suggest.

Players take turns drawing a rectangle each move, each rectangle using 4 of the dots as corners. Each rectangle must have a unique non-zero area and have unique corners (see below).

The sides of the rectangles may overlap those of previously drawn rectangles, but no corner should be the corner of a previously drawn rectangle (drawn by any player).
After drawing a rectangle, mark the 4 dots which are its corners with x's so that it is known which dots have been used already.

Also, after drawing a rectangle, write down in a (growing) list the area of this triangle (the area in terms of the "squares" of the original array of dots).
No rectangle may have the same area as any previously drawn rectangle (drawn by any player).

The last player able to successfully draw a rectangle using 4 previously-unused corners and having a unique area is the winner.
In other words, the player wins who moved just before the first player who THINKS he or she cannot move.

Thanks,
Leroy Quet

Sequence Ascension Solitaire Grid Game

2009-07-22T14:38:00.001-06:00

This is a game/challenge for one player. Start by drawing an n-by-n grid on paper. (n could be about 5 to 10, maybe.)

Next, either use a pre-existing integer sequence (such as {a(k)}, where a(n) = the number of divisors of n), or randomly pick the integers in the sequence.
If randomly picking the sequence, first write down n^2 numbers in a list before continuing to the next part of the game.
It might help to first write down the numbers even if you are using a preexisting sequence. Write down n^2 numbers, just in case they are needed.

Next, place the integers into the grid, starting at any square, and then placing each number -- in order by the order of the indexes (a(1), then a(2), then a(3)...etc), each number in any empty square horizontally, vertically, or diagonally adjacent to the last square filled with a number.
(a(k) is always next to a(k-1).)
Continue doing this until you cannot place any more integers (because there are no empty squares next to where you last put a number).

Next, starting at any square with a number in it, draw a path of connected line segments from square to adjacent square -- adjacent and in the direction of either vertical, horizontal, or diagonal -- such that each number drawn to is greater than or equal to the number in the previous square of the path.
(The numbers of the path never descend.)
The path must not visit any square more than once. But two diagonal segments of the path may cross.

Move until you can't move anymore. (The last square visited by the path will not be bordered by any unvisited square with a number >= the value in the last square.)

Your score is the number of squares your path visits.

Note: I realize that you could use the all-1's sequence, say, and score a perfect n^2 points each time, but that wouldn't be much fun.

As an easy challenge to myself I used the first 16 terms of the number-of-divisors sequence (1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,5)
and a 4-by-4 grid. I got a top score of 13 (in several ways).
Can you do better?

Thanks,
Leroy Quet

Dots To Primes Game

2009-07-22T14:37:00.001-06:00

This is a game for any plural number of players.

Materials: Blank pieces of paper, and a grid drawn on tracing paper. The horizontal rows of the grid are labeled in order 1, 2, 3, 4,..., such that there is one number per row.

Players take turns being the offense player.
At the beginning of a round, all of the players take turns placing dots on a blank piece of paper anywhere (anywhere where there isn't already a dot) within a large circle drawn on the paper. The size of the circle is the same each round.
A fixed total number of dots are drawn. This number is the same for all rounds.

After the dots are drawn, the offense player then rotates the tracing-paper grid in any way he/she desires, and places it over the circle of dots such that the circle is completely covered by the grid.

The offense player then reads the vertical positions of the dots from left to right -- relative to the grid. The offense player then reads the vertical positions of the dots from left to right -- relative to the grid. The offense player the forms the "first list" by writing down the numbers of the rows the dots fall into in order from the leftmost dot, relative to the grid, to the rightmost dot.

The offense player then forms a second list of partial sums of the first list.
The offense player starts this second list of numbers by first writing down the first number of the first list of numbers. He/she then adds the next number of the first list to the first number of the second (and of the first) list, and writes down the sum, then continues writing down all the partial sums, summed from left to right, until, finally, the last number in the second list is the sum of all the numbers in the first list.

Then the offense player circles all of the primes in the second list (the list of partial sums). The number of primes is the offense player's score for the round.

Then there is a new round with another offense player. Play continues until each player has been offense the same predetermined number of rounds.

Highest score wins.

Thanks,
Leroy Quet

Numerical Card Game

2009-06-30T09:11:00.001-06:00

This is a game for any plural number of players. It is suggested that there be at least 3 players.

Each player has an identical deck of m notecards, each card in each deck labeled with a different integer from 1 to m, where m is a positive integer large enough to make the game interesting.

Players pair off during the game, where every possible one of n*(n-1)/2 (n= number of players) pairings occurs, and all the pairing occur in some predetermined order.

Each player has two piles of cards: Their "moldy" pile (which starts out with zero cards in it), and their "fresh" pile (which starts out with every one of their cards in it). The cards in their fresh pile are all turned face-down, but can be looked at by the player who owns the deck.

On a move, the two paired-off players each pick any card they choose from their fresh piles, not revealing the cards until both cards are picked.
After both cards are picked, they are turned face-up.

One of three possibilities takes place:

1) If the one player's card number divides the number on the other player's card, then the player with the dividing number (not with the divided number) gets 2 points added to his/her score. The players then both put their cards in their own moldy pile (face-down), unless their card is a 1. If 1, then go to possibility #3.

2) If the cards' numbers are not coprime, and neither number is a divisor nor a multiple of the other card's number, then both players get 1 point added to their scores. The cards are then placed in the players' own moldy piles face-down.

3) If the cards' numbers are coprime, then the players exchange these two cards with each other. The players then each put the newly-gotten card in their own fresh pile, face-down.

(So, if one player has a 1, then he first gets 2 points added to his score, then must exchange the card with his opponent. Both players then put their cards back in their own fresh piles.)
(If both players pick the same number, then both players get 2 points added to their score; then their cards are put in the moldy piles; unless both cards are 1, in which case the cards are exchanged {if you feel this is necessary} and put in the players' fresh piles.)

A player stays in the game until his/her fresh pile is exhausted, or until he/she agrees to drop out.
(A player may agree to drop out if, say, it is clear that his remaining cards in his fresh pile are each coprime to every card remaining in every other player's fresh pile, and he has no 1. Then there would be no point in continuing.)

The game continues until there is one player left.

Highest score wins.

Thanks,
Leroy Quet

Lines By The Dots

2009-06-30T09:10:00.001-06:00

This is a game for any plural number of players.

Start with an array of n-by-n dots, where n is decided by the players amongst themselves. (There should be a higher n if there are more players.)

Players take turns moving. The first player to move starts on any dot.

Each player on each of his/her moves draws two lines:
On a player's move, he/she first draws a straight vertical line from {the dot where the last player left off [with a horizontal line, if this is not the first move of the game]} to {any dot in the same column as the dot the last player left off at}, making sure not to draw the line to or through any dot that has been drawn to previously by a line in the game.

Then the same player draws a straight horizontal line from where his/her vertical line ended to any dot in the same row, also making sure not to draw the line to or through any dot that has been drawn to previously by a line in the game.

The lines may not cross or coincide with another line. (This is obvious, since only "virgin" {not drawn to or from} dots may be drawn TO; although a dot need not be a virgin to be drawn FROM.)

If, and only if, a player cannot draw a line of the direction indicated by the rules, then he/she must draw a line (of the same direction that he/she was first unable to draw during his/her move) from any non-virgin dot to/through some of the virgin dots in the same row/column.

Scoring is as follows:

Before a player draws a line (vertical or horizontal), enumerate the number of non-virgin dots in the same column (if before drawing a vertical line) or same row (if before drawing a horizontal line). The player gets this number of dots added to his/her score.

Play continues until each dot is connected to by at least one line-segment. (ie. until each dot is not a virgin.)

Highest score wins.

Thanks,
Leroy Quet

Convoluted Coprimality Game

2009-05-29T10:14:00.000-06:00

Here is a game for any plural number of players.

Start by drawing an n-by-n grid on paper, where I suggest that n is >= 12.

Players take turns, going in a predetermined order (such as clockwise by how the players are seated).
The game starts when Player 1 places a 1 in the upper-left square of the grid.
Player 1 then places a 2 either in the square immediately to the right of the 1 or in the square immediately below the 1.
Then it is player 2's turn.

A "turn" is made up of a series of "moves". Only one player makes his/her moves during a single turn.

On the kth "turn", a player (whose turn it is to move) writes the numbers > than the (k-1)th prime and <= the kth prime in empty squares as follows:

On the kth turn, the integer j starts as (the {k-1}th prime)+1. The player continues moving until j equals the kth prime.
On the jth "move", a player places the number j in an empty square either immediately above, right of, below, or left of the square with a (j-1) in it. (The {k-1}th prime would have been placed in a square by the previous player to move.)
A number must be placed in a square bordered (above, right of, left of, below) by 2 or more squares that have already been filled in with numbers previously (filled in with numbers by any player). (So, the empty square to have the number j placed in it must be next to the square with the number (j-1) in it, plus the number j must be next to ONE OTHER square, at least, with a number already in it.)

BUT, if the square being filled in is in a row or column that is on the border of the grid, then the number j need only be next to ONE square that is already filled in (which is the square with the (j-1)).

When a player is forced to -- or does so by accident -- place an integer j in a square that is immediately next to (in the direction of above, below, right of, or left of) a square with a number that is NOT coprime to j, then that player is eliminated from competition.

Play continues until there is one player left, who is the winner.

If, during play, there are no empty squares where numbers can be placed, then the remaining players start again on a new empty grid, and j = 1 and k = 1 again.

Thanks,
Leroy Quet

Drat -- Grid Game

2009-05-13T10:05:00.001-06:00

Here is a game played on a n-by-n grid drawn on paper. The game is for 2 to 4 players. The size of the grid (n) should be relatively small as far as my games go, n = 4 to 6 for a 2-player game.

In the first phase of the game players take turns, each placing ANY integer from 1 to n^2 into any empty square of the grid on a turn.
(The same integer may be used more than once in the grid.)

In the second phase of the game, each player has a marker (each marker is small enough to fit inside a single square of the grid). Each player places his/her marker in a different corner square at the start of the second phase.

Players move in a predetermined order (such as clockwise by which corners the players start the second phase in).
The first player to move moves to any (above, below, right of, or left of, or DIAGONAL TO) adjacent square.

Players thereafter take turns moving their markers from square to adjacent square (in the directions of orthogonally or diagonally).

Say, the last player to move (whom we will call "player A") moved from a square numbered j to a square numbered k. Then on the next player's (Player B's) move, Player B MUST move, IF it is possible, to a square such that GCD(j,k) equals GCD(p,q), OR such that |j-k| = |p-q| -- where p = the number of the square Player B was on, and q = the number of the square Player B is moving to.

In other words, if it is possible, player B must move so that the numbers of his/her consecutively-moved-on squares have either the same greatest common divisor or same absolute difference as the last two consecutively-moved-on squares of Player A -- where player A is the player who moves just before player B (and where who exactly are players A and B changes each move -- player B always being the player currently moving).

If there are no such adjacent squares that have the same GCD or absolute difference, then player B may move to ANY adjacent square.

If a player is forced to move onto a square already occupied by another player's marker, then the SECOND player to occupy the square is removed from competition, and his/her marker is removed from the grid. (A player landing on an already occupied square is a 'drat', as in "Drat!".)

Play continues until there is one remaining player, who is then the winner.

Note: In the small number of trials where I played myself, it seemed that all games are either long or short, but never middle-lengthed.
Maybe some other math rules, besides GCD or absolute difference, would perhaps make this a more fun game.

Thanks,
Leroy Quet

PS: This game has been edited to allow for diagonal moves. Otherwise, with only orthogonal moves, one player may be unable to win no matter what.

Permutations Of +- Integers

2009-04-22T08:25:00.002-06:00

This is a game for two players, player 1 and player 2.

There are two lists of numbers being maintained, list A and list B.

Both lists start the game without any integers.

On move number n*, player 1 picks the sign (+ or -) for the integer, which has an absolute value of n, for list B.
And on move number n, player 2 picks the sign (+ or -) for the integer, which has an absolute value of n, for list A.

Then player 1 puts the integer (+ or - n), which had its sign picked by player 2, somewhere among the integers already in list A: either between integers already there, or at the beginning or end of the list of integers.
Then player 2 puts the integer (+ or - n), which had its sign picked by player 1, somewhere among the integers already in list B.

So, the each list, on move n, is a permutation of the integers
(+-1,+-2,+-3,...,+-n). On the nth move, the integers +-1, +-2, +-3,...+-(n-1) each remain in the same order relative to each other, but the +-n is inserted somewhere in each list.

*(A move consists of a series of steps, these being: Each player picking a sign for an integer, then each player placing an integer in a list.)

After a predetermined number of moves, the game is over. (Let the number of total moves be the positive integer m.)

Scoring is as follows:
Say, list A is the permutation (a(1),a(2),a(3),..,a(m)) of (+-1,+-2,+-3,...,+-m).
Let A(n) = sum{k=1 to n}a(k) be a partial sum of the first n terms of the permutation which is list A.
Player 2 (note: player 2) gets a point for every distinct k where |A(k)| = at least one |A(j)|, for 1<= j < k <= m, or where A(k) = 0.

Say, list B is the permutation (b(1),b(2),b(3),..,b(m)) of (+-1,+-2,+-3,...,+-m).
Let B(n) = sum{k=1 to n}b(k) be a partial sum of the first n terms of the permutation which is list B.
Player 1 (note: player 1) gets a point for every distinct k where |B(k)| = at least one |B(j)|, for 1<= j < k <= m, or where B(k) = 0.

Math question: How does the integer sequence {c(k)} start where c(n) = the number of permutations of (+ or - 1, + or - 2, + or - 3,...,+ or - n), where no partial sum of the first k terms of the sequence, 1 <= k <= n, equals any partial sum of the first j terms of the sequence, for all k and j where 0<= j < k <=n, no matter what the signs of the integers are?
(What are the permutations that gain 0 points for the player picking the signs, no matter how they pick the signs?)

Thanks,
Leroy Quet

Making Intersections

2009-04-15T14:44:00.000-06:00

This is a game for any plural number of players.

The game consists of a number of rounds, that number being a multiple of the number of players.
Each player takes turns being the offense player.

Before each round, draw an array of n-by-n dots (which correspond to the vertices of a grid of (n-1)-by-(n-1) squares).

On each round, all players take turns drawing straight vertical or horizontal line-segments from any dot to any other in the same row or column such that no line-segments cross any previously drawn segments and such that no line-segment coincides with any previously drawn line-segment. (But any line-segment may connect with a previously-drawn segment at a dot.)

The round ends when the players have drawn m line-segments, where m is a predetermined multiple of the number of players, and is picked by agreement among the players -- and m is the same for every round.
Note: m should be < n^2 - 2n + 4 = (n-1)^2 +3, which is the minimum number of segments needed to connect every dot to each of its orthogonal neighbors. (That this is the minimum number of segments is simply proved; and proving this may be a fun exercise for your amusement, if you don't want too hard a puzzle.)

After m line-segments are drawn, the offense player for the round receives a point for every dot connected to by 3 or 4 line-segments. (For the purpose of counting points, a single line-segment drawn by a player on a move and passing through -- and not terminating at-- a dot counts as 2 "line-segments" meeting at that dot.)

Players add up the points they received on the rounds that each was offense, and the highest total score wins.

Thanks,
Leroy Quet

PS: If this game is played on a rectangular array of j by k dots, the minimum number of straight horizontal and vertical line-segments (each of any length) needed to connect every dot to its orthogonal neighbors, without any line-segments crossing, is:
j*k - k - j + 4, or so I think.

Number Superposition Game

2009-04-02T08:35:00.001-06:00

Here is a simple game for any plural number of players.

This game is played on an m-by-m grid drawn on paper.

The first player to move places a 1 in any square of the grid.

Players thereafter take turns, each player placing the integer k in a square adjacent to (and in the direction of above, below, right of, or left of -- with restrictions {see below}) the square where the previous player to move wrote the integer (k-1). (k increases: 1,2,3,4,5,...)

A player may put the integer in an empty square OR may write the integer in a square that already has a single number in it -- but may not write a number in a square that already has two numbers written in it.

A player may also not write a number in the square previously written in by the player who moved before the last player to move.
(In other words: No U-turns. Any integer k cannot be placed in the same square as the integer (k-2).)

A player who writes the second number in a square gets {the absolute difference between the two numbers in the square} added to his/her score.

The game continues until no more moves can be made.

Highest score wins, of course.

Thanks,
Leroy Quet

Stepping By Divisors -- Grid Game

2009-03-17T09:14:00.001-06:00

Here is a game for any plural number of players. Start with an m-by-m grid drawn on paper. (I suggest that m be about 8 to 10 for beginners.) Draw the grid large enough so that two integers can be written in each square.

In the first phase of the game, players take turns writing the positive integers 1 to m^2 in order into the squares of the grid. One number is placed in any empty square of the grid on each move. (So, if there are 2 players, one player writes in the odd numbers, and the other player writes in the even numbers.)

Let the variable d (d for 'divisor') start the second phase of the game with a value of 1.

At the start of the second phase of the game, player 1 then writes the value of d, which is 1, alongside any number in the grid (in the same square as the number).

The players thereafter continue to take turns. On a move, a player chooses any square of the grid that has not yet had a second number written in it, but is adjacent to (in the direction of above, below, right of, or left of) any square that has had a second number written in it.
He/she then writes down in the square (with one number) any* positive divisor of the number in that square.
The variable d then becomes that divisor.
* The value of d, however, must change each move. The same divisor number cannot be written in two squares on two consecutive moves.

The absolute value of the difference between the older recent value of d (the divisor written by the previous player to move) and the new value of d (the divisor written by the current player moving) is then added to the currently moving player's score.
Note: The goal of the game is to get the LOWEST score. So, it is advantageous to change the value of d by as little as possible on a move. (Changing the value of d by 1 is the best a player can hope for on a move.)

The game continues until each square has exactly two numbers in it.
(So, there are a total of m^2 moves in the first phase of the game, and m^2 moves in the second phase of the game.)

As I said before, the player with the lowest score wins.

I would suggest that the divisor numbers (the values of d) be written smaller than the numbers written during phase 1 of the game, or be written in another color than the first numbers placed in the squares.

PS: The only problem I can see with this game is if the last square to get a second number has a 1 in it, and the previous (next to last) player to move placed a 1 as the second (divisor) number in some square. (This is a problem because d must change each move.)
Then, in that case, the second phase of the game ends after m^2 - 1 moves.

Thanks,
Leroy Quet

Permutations Of Divisors

2009-03-14T07:54:00.000-06:00

This is a game for any number of players. (Gosh darn, no grids this time.)

This game consists of a number of rounds, where the total number of rounds is predetermined and is a multiple of the number of players.

Players take turns choosing integers, one integer per player per round.
On a round, the player whose turn it is to chose picks any positive integer that has not yet been chosen in the game to be placed at the end of a growing list of integers.
(So, after the player picks a number during round n, there are then exactly n integers in the list.)

Say this list (the "divisor list") is (d(1),d(2),...,d(n)).
On the nth round, after the nth term is appended to the divisor list, each player (by themselves and in secret) then tries to come up with a positive integer m such that, if (d'(1),d'(2),...,d'(n)) is a player's permutation of the divisor list, then
d'(j) divides (m+j-1) for all j where 1 <= j <= n, if such a permutation exists.
In any case, each player tries to find a permutation of the divisor list where as few terms as possible are not in their same position as they are in the original divisor list, and where as few of the members of the divisor list as possible do not divide the numbers in the "multiple list" they are paired with (where the "multiple list" is the list of consecutive integers from m to m+n-1).

So, in other words, a player's score on a round begins as the number of j's where d'(j) = d(j), where {d'(j)} is the player's own permutation of the divisor list, and where d'(j) divides m+j-1. Then, to get the player's score for that round, subtract the number of j's where d'(j) does not divide (m+j-1) (1 <= j <= n).

A player grand score is the sum of his/her scores from each round.

The player with the highest grand score wins.

For example, let us say that n = 6. And the divisor list looks like this:
1, 2, 4, 3, 17, 5.

A player then chooses this multiple list:
13, 14, 15, 16, 17, 18.

That same player then choses this permutation of the divisor list:
1, 2, 3, 4, 17, 5.

Only the 3 and the 4 are out of place. And only the 5 does not divide its respective integer in the multiple list.

So this player for this round gets
3 - 1 = 2 points, because only 3 integers in his/her divisor list permutation (Those 3 integers are 1,2,17) both divide their respective integers in the multiple list and are not out of order from their positions in the original divisor list.

Thanks,
Leroy Quet

Connecting Graphs

2009-03-07T16:24:00.000-07:00

This is a game for any number of players.

Start with an n-by-n grid lightly drawn on paper. Or perhaps a square array of dots (representing a grid's vertices) would be better.

Players take turns connecting any ADJACENT pair of dots/vertices with a straight line-segment, one segment each move. Only pairs of dots that have not been previously connected may be connected on any move. Although any particular dot/vertex can have multiple line-segments drawn to it.

Line-segments may be vertical or horizontal. In a variation of the game, diagonally adjacent vertices/dots may be connected as well, as long as no line-segments cross. (In the variation, each line-segment is drawn either N, NE, E, SE, S, SW, W, or NW.)

Borrowing a term from graph-theory, a "graph" of connected line-segments (each line-segment drawn during the game's play) is a "connected graph" if one can trace along the line-segments from any vertex of the graph to any other vertex (possibly, but not necessarily, connecting each vertex to any other in multiple ways).

Whenever 2 distinct connected graphs are combined into 1 connected graph by a line-segment, the player drawing that line-segment gets a point.

No points are obtained for starting a new connected graph, for extending a single connected graph (in a way that does not connect to another connected graph), or for connecting a connected graph back to itself.

The game is over as soon as there is exactly one connected graph, no more, of line-segments drawn on the grid. The initial single connected graph, formed when the first move of the game is made, does not end the game. (Otherwise, what a dumb game this would be!)

Highest score wins.

How does allowing diagonally drawn line-segments alter the game, I wonder?

This game sounds familiar too. Where have I stolen the idea from?

Thanks,
Leroy Quet

Words, Letters, & Logic

2009-02-26T12:48:00.000-07:00

This is a word game inspired by the mathematical game at the link below:
(But this game should be more fun for the anti-math crowd.)

http://gamesconceived.blogspot.com/2009/02/arranging-numbers-by-rules-game-also.html

This game, which is for 2 players, is sort of like a cross between Scrabble and Sudoku. Sort of.

The game is divided into several phases.
In the first phase the players take turns writing letters into the squares of a 4-by-4 grid (or a 5-by-5 grid for more advanced players).
Each letter should appear at most once in the grid.

So, for example, we could have this grid:

L S R Q
K C G P
A D B H
M E I N

Then, in the next phase the players take turns coming up with a list of words associated with each letter of the grid. Let the letter associated with a word be a "grid-letter" (because the letter appears in the grid). A word or phrase (that may be almost nonsense, if no shorter words can be thought of) must contain all of the letters in the squares immediately adjacent (in the directions of above, below, left of, and right of) to the word's grid-letter, but the word need not contain the grid-letter itself.

The list of words, each word written next to its grid-letter, is ordered by the grid-letters in alphabetical order.

So, in my example we can have the list of words (written right of their grid-letters):

A: mocked
B: blighted
C: dark gods (Notice that this is an arbitrary-sounding phrase.)
D: backed
E: mined
G: backpacker
H: pinto bean
I: bean
K: lack
L: sky
M: rake
N: high
P: quag hole (Another arbitrary phrase. A quag is a marshy place.)
Q: pray
R: quagmires
S: clear

For example, the letter A in the grid is next to D, K, and M. And the word "mocked" contains these letters.
(Note: Words with lots of letters are more likely to make the game easier for both players. Words with fewer letters are more likely to make the game harder for both players.)

In the next phase the players each draw their own empty 4-by-4 (or 5-by-5) grid.
Then each player writes any one letter that occurs in the original grid into any square of their OPPONENT'S grid.

Then the original grid of letters is hidden. (So the list of words should be drawn on a different piece of paper than the original grid of letters.)

In the final phase the players try to each fill their own grid, given the letter written in their grid by their opponent, with the same letters that were in the original grid (one letter per square of the grid), never repeating a letter (including the one letter written by the player's opponent in the player's grid), such that all letters adjacent (in the directions of above, below, right of, and left of) to a letter in the grid occur in the word associated with that grid-letter.

Remember that if two letters are adjacent (say, letter 1 and letter 2), then letter 1 must be in letter 2's word AND letter 2 must be in letter 1's word.

The players each try to fill in as many squares as they can under the rules.
Their score is the number of squares they correctly fill in with letters.
(Note: Not all grids can be filled in completely. It depends on where a player's opponent places the first letter in the player's grid.)

If a player makes a mistake (a letter doesn't appear in one of its adjacent letter's words, a letter is written in a player's grid that wasn't in the original grid, or a letter occurs more than once in a player's grid), than that player forfeits.

If neither player forfeits, then the winner is the player who filled in the most number of squares in their own grid.
(Ties possible.)

Back to the example, here is a player's grid with an E put in the lower left square by the player's opponent:
(* is an empty square.)

Q P H B
R * * *
M A K L
E D C S

13 points.

(Note: I couldn't put a G in the empty square at position (2,2) because there is no G in "mocked".)

Thanks,
Leroy Quet

Arranging Numbers By Rules -- A Game Also A Puzzle

2009-02-22T07:25:00.002-07:00

This seems like it would be a fun game.

This game is for 2 players. Start by drawing an n-by-n grid on a piece of paper, where n is at least 4 or 5 (but not too massive). I suggest that n be even (to make this game fair for both players).

First the players take turns placing the integers 1 through n^2 into the grid's squares so that there ends up being exactly one integer in each square of the grid.
Player 1 places the odd integers in the grid's squares, and player 2 places the even integers.

Then the players take turns making up rules or classifications, one rule per each integer from 1 to n^2, where each rule defines a class of integers which includes all the integers immediately adjacent (in the directions of above, below, left of, and right of) to the integer which matches the number of the rule.
(The rule need not match the number of the rule itself.)
In other words, say that we are concerned with the rule defining the neighbors of the integer 3 in the grid. Left of the 3 happens to be, in this example, a 5. Above the 3 happens to be a 9. Below the 3 is a 2. And right of the 3 is a 17.
So, the neighbors of the 3 are 5, 9, 2, 17. There are of course an infinite number of classes that these number fall into. But one of the classes is (2^k + 1), since all 4 of the integers are 1 more than a power of 2. So rule #3 could be "Numbers of the form (2^k +1), k >= 0".

So, player 1 makes up the rules for the neighbors of the odd integers. And player 2 makes up the rules for the neighbors of the even integers.

I encourage players to be creative when coming up with rules. Yes, a rule could look like: "One of these integers: 2,6,5,9", or on the other extreme: "Any integer at all". But making a rule too broad or too narrow affects both players equally.

Next, after the rules are constructed, each player draws an empty n-by-n grid for themselves. Each player then places into any square of their opponent's grid any integer from 1 to n^2.

Then, each player tries to fill in the remaining squares of his/her own grid so that, given the integer her/his opponent already placed in her/his grid, each integer's immediate neighbors (in the direction of above, below, right of, left of)
follows the corresponding rule for that integer. The original grid of numbers is hidden while the players each try to solve the puzzle.

Remember that if two integers, j and k, are adjacent, then not only does j have to follow rule #k, but k has to follow rule #j as well.

A player's score is the number of squares he/she fills in successfully. If a player makes a mistake (a number doesn't follow the rule for the number it is adjacent to, or a specific number appears more than once in a player's grid, or a number in the player's grid is not an integer >= 1 and <= n^2), then the player forfeits.

*****
Update:
I also should make an addition to how the game is played.
Rules such as rule 7 below, where other squares' values have something to do with the rule, can be problematic if not all the relevant squares are filled in.
So, if rule #m says that the values of the neighbors of the m must depend on other square's values in some way (the other squares which rule #m depends upon we call S), then the squares of S must all be filled with numbers, IF any of the neighbors of square m are filled, or else the player forfeits.

In my example, all the numbers in the same row as the 7 must be filled, if any of the numbers adjacent to the 7 are filled, or else I would automatically lose to my opponent (unless we both forfeited for any reason).

*****

Here is a sample 4-by-4 grid with rules.

01 04 09 16
02 03 08 15
05 06 07 14
10 11 12 13

1: Power of 2.
2: Prime power.
3: Even.
4: Power of 3.
5: Squarefree.
6: Prime.
7: Not coprime with the sum of the integers (including the 7) in the same row as the 7.
8: Odd.
9: Power of 2.
10: One less or one more than a triangular number.
11: Prime - 1.
12: Prime.
13: Between 10 and 15 (inclusive).
14: Divides 14 or is coprime to it.
15: Squarefree integer + 1.
16: Multiple of 3.

Let us say, without me trying to actually solve the puzzle, that the player's opponent puts a 3 in the lower left square of an empty 4-by-4 grid. How many integers can be put in this grid correctly under the rules?

Note: Any integer (from 1 to n^2, not occurring anywhere else in a player's grid) can occur anywhere in a player's grid where there would be no integers in the neighboring squares immediately above, below, right of, or left of it, of course. (No rules are violated here.)

Thanks,
Leroy Quet

Primes, Moves, & Motions

2009-02-17T07:39:00.001-07:00

This is a game for 2 players. It is played on an n-by-n grid, where n is at least 8 or higher, I suggest.

Player 1 starts the game by placing a 1 in any square of the grid.

The game consists of "moves" alternately taken by each player. Each move is made up of a series of "motions", where a single player makes all of the motions in any particular move.

A player on move n (where player 1 placing the 1 any square is move #1 and is motion #1) makes motions p(n-1) through p(n)-1 (for moves n>=2), where p(n) is the nth prime.
Player 1 makes the odd numbered moves, while player 2 makes the even numbered moves.

On MOTION m, a player places the number m in any EMPTY square that is adjacent to the square with a (m-1) in it (which was placed in the (m-1) square by either player), such that:

*If m is an even composite, the player places m immediately either left of or right of the square with an (m-1) in it.
*If m is an odd composite, the player places m immediately either above or below the square with an (m-1) in it.
* If m is a prime (ie. If this is the first motion of a player's move), then the player can place m in the square that is immediately either above, below, right of, left of, or diagonal to the square with the (m-1) in it.

The last player that can make a motion loses.

Variation: The first player that cannot make a motion loses.
(The difference between the original version and the variation is simply that in the original version, if a player places a p-1, where p is a prime, but the other player can't place a p, then the player who placed the p-1 loses. In the variation, the player who cannot place a p loses.)
I leave it up to players to decide amongst themselves which version they prefer.

Thanks,
Leroy Quet

Draw Lines And Shade Sections

2009-02-12T09:33:00.000-07:00

(I have posted other games before where you draw lines, then shade in sections bordered by the lines. But I can't think of a better name for this game.)

This game is for any plural number of players.

Start with an n-by-n grid lightly drawn on paper.

Players take turns drawing horizontal and vertical line segments, each segment being one grid-square side in length, from grid-vertex to adjacent vertex along the lightly drawn lines of the grid.
Player 1 starts the game by drawing a line segment from any vertex to adjacent vertex. Players each, thereafter, draw a line segment from where the last line segment left off. Darker line segments must not be drawn where other darker line segments were previously drawn. And the continuous path of line-segments must not cross itself. Yet, the path may be drawn to any vertex more than once.
This part of the game continues until the path cannot be drawn anymore.
(If all players want an interesting game, they should probably try not to lead the path into a situation where it prematurely ends.)

After the darker path of line-segments is complete, the players then take turns filling in un-shaded-in sections of the grid, one section per move. By "section", I mean a polygon bounded by the darker line-segments, or by vertexes where different parts of the path come together, or by the perimeter of the grid. (You can draw darker line segments along the border of the grid. But as far as the sections and the border of the grid are concerned, whether a particular segment of the border of the grid was darkened in or not does not matter.)

As soon as a player is forced to fill in a section bordering (along a line) another filled in section, or accidently does so, then that player is removed from play.
(Two filled in sections may border at a vertex without removing a player.)

Play continues until there is one player left, who then is the winner.

By the way, I suggest that when two parts of the path come together at a single vertex, then the path should be drawn so it is clear that there is no gap between the parts of the path. Otherwise, players may think that two of the sections that meet at that vertex are only one section (with a choke-point).

Thanks,
Leroy Quet

PS: Don't get confused between "sections" and "segments".

Squares Overlapping/ Subdividing Into Primes

2009-02-05T08:51:00.002-07:00

Here is a game for any number of players.

Play a number of rounds, where the number of rounds is a multiple of the number of players. Each player plays the same number of rounds as offense.

Start each round with with an n-by-n grid lightly drawn on paper, where n is at least 15 or more and is finite. (And n is the same value for all rounds.)

On a round the players take turns boldly drawing the perimeters of squares along the lines of the grid. The edges of the squares may overlap. But each player must darken in at least some segment(s) of grid-lines that have not yet been part of any edge of any previously drawn square.

After a predetermined number of moves (the same number for all rounds), then the offense player tries to find the largest CONTIGUOUS collection of regions bordered by the bold lines that have a total of a prime number of grid-squares in them.
(The regions can be of any shape, and may be made by adding squares together or by taking smaller squares out of larger regions, for example.)

The offense player gets this prime added to their score.

After all rounds are played, the player with the highest total score wins.

Thanks,
Leroy Quet

Polygonix

2009-01-28T09:19:00.001-07:00

Here is a game for two players. It can be played on a grid, but this isn't necessary. (Although playing on a bounded grid eliminates a lot of the ambiguity that may occur about the positions of dots.)

The players take turns being the defense player and the offense player.

The defense player starts a round by drawing m (m is a positive integer determined ahead of time) dots on a blank piece of paper. If you are using a grid, the defense player draws the dots at some of the intersections of the grid-lines.

Then the offense player connects pairs of dots with straight line-segments. The offense player continues to do this until all of the dots are each on the perimeter of at least one CONVEX polygon, and none of the dots are on the perimeter of any concave polygons or on the perimeter of any polygon that is not simply-connected.

(By "polygon", here I mean a region bounded completely by line-segments with no line-segments through its interior {except possibly grid-lines}.)

The offense player gets a point for each polygon.

Players play an even number of rounds, switching who is offense and who is defense, then add their scores.

The LOWEST score wins. (So, players try to minimize the number of convex polygons they draw when they play offense.)

Thanks,
Leroy Quet

Intersections Of Rectangular Loops

2009-01-20T12:59:00.001-07:00

Here is a game for 2 player, each player with a colored pen/pencil of a color different than her/his opponent's pen/pencil. This game is played on an n-by-n grid drawn on paper. (I will call the two players, for convenience, player-yellow and player-purple.)

Players take turns filling in empty squares of the grid, one square filled on each move, such that each player fills in exactly n squares during play (2n squares filled in by both players together) and such that there are exactly 2 squares (no fewer, no more) in each row of the grid and exactly 2 squares in each column.

After the squares are filled in, someone draws a straight horizontal line segment between the centers of each pair of squares in the same row, and draws a vertical line-segment between each pair of squares in the same column. You then should have 1 or more closed curves consisting of straight line-segments and 90-degree turns.

Scoring is as follows:

What matters in this game are the intersections of the line-segments. (Which closed-curves the line-segments of an intersection belong to is unimportant in this game. A closed curve may even intersect itself, of course.)

Call the 4 squares that are in the same row and column as an intersection of 2 perpendicular line-segments the 4 "extremities" of the intersection.
Call the pair of extremities that are both along the vertical line-segment of the intersection, or are both along the horizontal line-segment of the intersection, a pair of "opposing extremities".

Look at each intersection. (I suggest putting a circle in the square with each intersection, just to make the intersections easier to see.)
For every intersection where: {{the horizontal opposing extremities are of the same color} and {the vertical opposing extremities are of the same color}} or {{the horizontal opposing extremities are of differing colors} and {the vertical opposing colors are of differing colors}}, player-yellow gets a point.

On the other hand, for every intersection where one pair of opposing extremities consists of 2 squares of the same color and the other pair of opposing extremities consists of 2 squares of differing color, then player-purple gets a point.

Here is a much simpler way, probably, to figure out who gets a point at any intersection.

Count the number of the intersection's extremities colored by player-yellow or count the number of extremities colored by player-purple. If the number of an intersection's extremities filled by either one player is even, then player-yellow gets a point for that intersection. If the number of an intersection's extremities filled by either one player is odd, then player-purple gets a point for the intersection.

Highest score wins.

Thanks,
Leroy Quet

Doppel-game

2009-01-16T07:21:00.000-07:00

(Title is taken from "doppelganger".)

This game seems to be familiar. And the rules are simple. So, maybe, I might have already posted a game with similar rules. Or a similar game might have been invented by someone else. (Actually, I could include this disclaimer with almost any of my games.)

Here is a game for 2 players played on an n-by-n grid.

First, fill in any one randomly chosen square of the grid.

Players then take turns filling in empty squares of the grid, one square per move, such that any square being filled in is immediately next to -- and in the direction of above, below, right of, or left of -- any square anywhere on the grid that has already been filled in (by either player).

({}'s added for clarity below.)
Say, a player (player A) fills in a square that is immediately next to -- and in the direction of above, below, right of, or left of -- the square that same player (player A) filled in in their last move. Then let the direction from {the previously filled-in square (from the previous move of the same player, player A)} to {the newly filled-in square} be the direction d.
If the direction from {ANY filled-in square immediately next to {the square the other player (player B) last filled in}} to {the square the other player (player B) last filled in} is d, then player A gets a point.

No point is obtained if player A doesn't fill in a square immediately next to the square previously filled in by the same player (player A) or if the direction from {player A's previously filled in square} to {the current filled in square on player A's move} does not equal {a direction from any filled in square (immediately adjacent to the last square filled in by player B)} to {the last square filled in by player B}.

Got that?...
(I said these rules were simple!...Ha! -- Well, the rules ARE simple, once you figure out what they are!)

Play continues until all the squares of the grid are filled in.

The player with the most points wins.

Clarification:
First of all, player A refers to either player, but is the player currently moving.

Call 3 consecutive moves "move m", "move (m+1)", and "move (m+2)".
Player A made moves m and m+2, and player B made move m+1.

The direction from the square filled in on move m to the square filled in on move m+2, if those two square are immediately adjacent (in the direction of up, down, left, or right), is direction d.

Let "square (m+1)" be the square filled in by player B on move (m+1). If the direction from {ANY filled-in square immediately next to square (m+1)} to {square (m+1) itself} is direction d, then player A gets a point on move (m+2).

Any comments?

Thanks,
Leroy Quet

Diagon -- The Game

2009-01-07T09:49:00.001-07:00

Here is a game for any plural number of players.

Start with an n-by-n grid drawn on paper, where n is an odd positive integer.

Fill in the 4 corner squares of the grid.

Players take turns filling in grid-squares, one square per move. On a move a player fills in any empty square that is adjacent to and immediately above, below, right-of, or left-of any filled-in square.

A player gets a point when that player fills in a square such that that square is in a 2-by-2 group of squares where two diagonally adjacent squares are filled in (including the square just filled in) and the other two diagonally-adjacent squares are empty (empty just after the move when the point is scored).

For instance, a point is scored if we have a 2-by-2 group of squares that looks like this:
* o
o *

or this:
o *
* o

(o = empty square, * = filled in square, where one of the filled in squares is the square just filled in by the scoring player.)

The game continues until there is no possibility that any more points can be scored.

Highest score wins.

--

If during the game we have a situation like so, say:
o *
* o
o *
(newly filled in square is the middle filled-in square), then how many points has the player scored? One or two?I leave how many points that can be scored on any single move, one at most or more, be up to the players to agree to among themselves.

--

Question: Is there a simple way for one player, say the first or second to move in a 2-person game, to always win? (If there is, I probably should edit this game to eliminate the possibility of using the simple strategy.)

Thanks,
Leroy Quet

Uncooked Spaghetti And Square Meatballs

2008-12-23T14:32:00.000-07:00

(This game sounds familiar. Is it original?)

This is a game for 2 players.

Start with a carefully drawn n-by-n grid.

The players take turns completely filling in a total of n squares of the grid. (So, the first player to move fills in ceiling(n/2) squares, and the second player to move fills in floor(n/2) squares.)

After the squares are filled in, then the second player who filled in the squares is the first player to move in the next phase of the game.

The players take turns. On each turn a player draws a straight line from any empty vertex on the edge of the grid (where a grid-line meets the grid's perimeter) to any other empty vertex on any other edge of the grid.
By "empty" vertex, I mean a vertex that has not yet had a line drawn to it or from it in this phase of the game.

No line may pass through any filled-in square. But a line may touch a filled-in square (along an edge or touching at a corner).

(Also, lines may be vertical or horizontal. For this reason, I suggest that the grid be lightly drawn.)

Every time a line passes through a previously drawn line (previously drawn by either player in the second phase of the game) then the player's OPPONENT gets a point for each line crossed by the player's line.

Players move until there are no more possible lines that can be drawn under the rules.
(If a player claims that he/she cannot move any more, then the player's opponent may challenge this assertion and find, if possible, a path the player's line can indeed follow.)

Highest score wins.

Thanks,
Leroy Quet

PS: See the post (to my blog "Amorphous Trapezoid") about games-related poetry at:
http://prism-of-spirals.blogspot.com/2008/12/blog-post_21.html

Palindromic Card Game

2008-12-23T14:29:00.000-07:00

This is a card game for 2 players. (Although this game doesn't technically require cards, using them makes the game easier to play.)

Start with 2n cards, n red and n black cards. (You can play with a standard deck {no jokers}, letting n be 26, and all spades and clubs are black, all hearts and diamonds are red. All that matters in this game is the colors of the cards' suits.)

Deal n cards to each of the players.

Players arrange their n cards in any order in a row, face up.
(One row of cards per player.)

After the cards are arranged, each player then tries to find as many distinct palindromes (symmetric patterns of of redness and blackness) within their opponent's row of cards, where each palindrome starts and ends with a red card. By DISTINCT palindrome I mean that each particular arrangement of reds and blacks counts only once. (Also, different distinct palindromes may share some of the same cards. And a palindrome may consist of exactly one red card.)

For example, if we have an n of 12 and we have the following row:

R B B R R B B B R B B R

Then the palindrome (R B B R) would be counted once, even though it occurs twice in the row.
(The palindrome (BBRBB) would not count at all because it starts and ends with black cards.)

A player gets this many points:

(number of red cards in the player's row) - (number of DISTINCT palindromes found by the player's opponent in the player's row).

(This score will always be 0 or higher.)

Highest score wins.

Thanks,
Leroy Quet

PS: See the post (to my blog "Amorphous Trapezoid") about games-related poetry at:
http://prism-of-spirals.blogspot.com/2008/12/blog-post_21.html

"Maze" Of Polygonal Sections, Game

2008-12-12T07:45:00.000-07:00

This game has elements in common with an earlier game of mine, Slice And Fill.
See:
http://gamesconceived.blogspot.com/2008/09/slice-and-fill.html

This game will work with any number of players.

Start with an n-by-n grid lightly and carefully drawn on paper.

Darken in the 4 grid-lines that form the square boundary of the grid. (All of the vertexes along the grid's edge are thereafter each considered to be drawn-to by a line-segment.)

Players take turns drawing straight line-segments, one segment per move, each segment drawn from any vertex of the grid that has a line segment passing through it or terminating at it, to any vertex that touches no line-segments, such that the line segments don't cross any others or coincide with any others.
(Any number of segments may be drawn FROM any single vertex. Line-segments may be diagonal and of any slope. Each line-segment may pass through any number of vertices. But, I repeat, line segments must each be drawn from a vertex of the grid to another vertex of the grid, not from an intersection of a line-segment and a grid-line if that intersection is not a vertex of the grid.)
The first line-segment (after the perimeter of the grid is filled in) is drawn from a vertex along the edge of the grid, of course.

When all vertexes of the grid are touching line-segments, we have a maze (without an entrance or exit), and then the next phase of the game begins.

One player starts the second phase by filling in any "section" of the subdivided grid. A section is defined by* the lines of the grid and/or by straight line-segments drawn by players (a section may be a square, or it may be a polygon which is a subset of a square).
*[By "where the section is defined by...", I mean "where the section is BORDERED by" the lines of the grid and/or by straight line-segments drawn by players. There are no internal line-segments within any given "section".]

Then the players take turns filling in, if possible, any UNFILLED section that is immediately adjacent to the previously filled in section (filled in by another player) and that is not separated from the previously filled in section by a line-segment drawn in the earlier phase of the game. (So, consecutively filled sections must not only be adjacent, but must be in the same "corridor" of the maze.)
If a section can be filled in under the rules, then a section must be filled in.

If, however, a section cannot be filled in by a player (either because it is surrounded by already filled in sections, or it is at one of the maze's dead-ends), then the previous player to move gets a point. The player who cannot fill in a section under the rules above then fills in any unfilled section (so as to start a new string of filled in sections).

The game continues until all sections are filled in.

The player with the greatest number of points wins.

Thanks,
Leroy Quet

Co-Compositeness (a game)

2008-12-02T08:42:00.000-07:00

This is a game for any number of players. Start with an n-by-n grid, where n is larger if there are more players. (I suggest an n of at least 16 if there are 2 players.)

The first player to move places a 1 in any of the grid's squares.

Players take turns placing numbers in the grid's squares as follows:

*Each player places in a grid-square the next higher odd integer than the (odd) integer previously put in a square by the previous player to move. (So, let m be the number of total moves made by all the players so far; then the next player to move places a {2m+1} in the next square.)

*Players place the (odd) integer in a square that is immediately adjacent (in any of the 8 directions of: above, below, left of, right of, or diagonally) to the square the previous (odd) number was last put inside.

*Each integer is either placed in an empty square or in a square that already contains just one number that is NOT coprime to the integer the player is now placing in the square. There may be no more than 2 integers in any one square.

Scoring:
Every time a player places an integer in a square with an integer already in it, then that player gets a point. (Any pair of integers in the same square must be "co-composite", ie non-coprime.)

Players continue filling in the squares until a player cannot move anywhere. (If a player can move, then the player must move.) Then the game is over.

Highest score wins.

(Note: Part of the strategy of this game might be for a player to try to force an early ending to the game if that player has the highest score so far.)

Thanks,
Leroy Quet

Interconnected Subdivided

2008-11-24T11:16:00.002-07:00

This game has been removed because it is dumb.

f(g(x)) = g(f(x)) (a game)

2008-11-18T10:02:00.001-07:00

This game is for 2 players.
First, the players each secretly come up with two mathematical functions (two functions per player), f(x) and g(x).
The functions, f(x) and g(x), are defined and real for all real x, and they are continuous.

Next, the players take turns each being the "proposer" and the "solver".

First in a round the proposer reveals his/her functions.

The solver tries to determine (within a given time period) whether
f(g(x)) = g(f(x))
either for (1 of 3 choices) no real x's, a finite number of real x's, or an infinite number of real x's.

If the solver can't do this in a specific finite period of time, then the proposer gets a point. Otherwise the solver gets a point.

The players then switch who is the proposer and who is the solver.

(So, with only the two rounds, this is a low-scoring game.)

Example: f(x) = sin(x). g(x) = e^(x+1).
So, f(g(x)) = sin(e^(x+1)). g(f(x)) = e^(sin(x)+1).

The number of x's where sin(e^(x+1)) = e^(sin(x)+1) is precisely the number of pairs of integers (m,n) such that:
e^(1- pi/2 +2*n*pi) = pi/2 + 2*pi*m (I think).

I don't personally know if this equation is solvable at all for any integers m and n, let alone if there are a finite number of pairs (m,n) or an infinite number of pairs of integers m and n. So, if I was the solver and if the proposer proposed f(x) = sin(x) and g(x) = e^(x+1), then the proposer would get a point that round, and I would get no points that round.

Grid-Squiggly Game

2008-11-11T15:01:00.001-07:00

This game is played on an n-by-n grid (n by n squares, {n+1} by {n+1} lines) that is lightly drawn on paper. The game is for 2 or more players.

First, players each secretly pick a positive integer m between 1 and n^2. (See below.)

The first player to move draws a line segment, one grid-square in length, along any vertical or horizontal line of the grid. (But don't draw along the border of the grid.)

Players then take turns drawing a line-segment each turn, where the line-segment is one grid-square in length, and goes from any vertex with a line-segment drawn to it (by any player) to any adjacent vertex that does not yet have a line segment drawn to it. (The drawn-to vertex is immediately above, below, right of, left of the drawn-from vertex.)
No line segments go along the border of the grid, but line segments can connect to vertices along the border of the grid.

Players continue to draw segments until there is no place they can draw them. (A total of n^2 + 2n - 4 segments will be drawn.)

Next, with a pencil of a color different that their opponents' pencil colors, each player takes turns (completely) filling in sections of the grid, one section each move. Each "section" is bounded by the lines the players drew and by the border of the grid.

When the whole grid has been colored in, count the number of squares filled in by each player.

Players then reveal the numbers (m) they picked at the game's beginning.
The player whose number of squares filled in is closest to the number they picked at the game's beginning (m) wins.

Thanks,
Leroy Quet

Within The Curve

2008-10-28T16:07:00.001-06:00

Here is a game for any number of players.

Each player has m (n-by-n) grids, where m is the number of players. (So there are m^2 grids all together.)
I suggest an n of at least 12.

On one of their grids each player secretly draws a closed non-self-intersecting curve. (The curve is bounded within the n-by-n grid.) Each player's curve does not go through any intersections of the grid-lines.

Next, on one of each of the other player's blank n-by-n grids each player copies his/her curve over.
The copies of each curve must go through the same respective squares of each grid as the original curve did.
So, there are m copies each of m curves, each player in possession of one copy of each curve.

Next, secretly and simultaneously, each player fills in the squares each curve goes through on any particular grid with 1,2,3,...., the integers placed in order and next to each other along the curve. The numbers can start anywhere along a curve, and can go either clockwise or counterclockwise.

Next, each player secretly fills in the squares within each curve's interior with 1,2,3,..., the numbers placed in order, each number placed in any empty interior square such that all other numbers (including possibly numbers along the curve) above, right of, left of, or below the number are coprime to that number.

(Any pair of adjacent numbers that are both in squares a curve passes through don't have to be coprime. Only interior numbers have to be coprime to adjacent numbers along the curve, or to adjacent numbers that are also on the curve's interior.)

Players continue to fill the interior of each curve with numbers until the players can't fill in any more numbers under the rules.

When each player has filled in each curve as far as they can, the score for each player is the sum of the top numbers in the interior squares of each of the m curves the player filled (partially) in.

Highest score wins.

Players may check their opponents' grids after the game is over to make sure that all applicable pairs of adjacent numbers are actually coprime. If a player made a mistake, that player automatically loses the game.

Thanks,
Leroy Quet

Polygons In Permutation Grid

2008-10-14T16:20:00.001-06:00

Here is a game for 2 players.

Start with an n-by-n grid (n-lines by n-lines, or n-1 squares by n-1 squares). I suggest that n be at least 10.

First, players take turns placing a total of n dots at intersections of the grid.
Each dot is placed at an intersection of any two lines that do not have any other dots on either of them.
So, after n dots are placed on the grid, the dots represent a permutation of (1,2,3,...n).

Reading the dots from top to bottom, let the dot on the mth horizontal line be p(m).
Reading the dots from left to right, let the dot on the mth vertical line be q(m).

Draw a straight line-segment from p(m) to p(m+1) for all m where 1<= m <= n-1.
Draw a straight line-segment from q(m) to q(m+1) for all m where 1<= m <= n-1.

Player 1 gets a point for every triangle that is formed by the line-segments.
Player 2 gets a point for every non-triangle (4 or more sides) that is formed by the line-segments.
Only polygons completely bounded by parts of line-segments (not counting the grid's lines) score any points.

For a triangle or non-triangle to score a point, the polygon must not be subdivided by any line-segments (but may be subdivided by grid-lines).

I suspect that there is a bias either towards player 1 or player 2. So, play an even number of rounds with the same-sized grids, each player playing player 1 and player 2 an equal number of times, and add up each player's score to get the players' grand total scores.

Highest grand total score wins.

Thanks,
Leroy Quet

Cross, Don't Cross Circle Game

2008-10-08T10:55:00.001-06:00

Here is a game for 2 players.

As in many of my games, the players play an even number of rounds, half of the rounds where one player is offense and the other player is defense, and the other half of the rounds with the players switching who is defense and offense, and then the players adding up their scores for their grand total scores.

A round starts with a carefully drawn circle on paper. (No grids this time. Sorry.)

The defense player starts the game by drawing m (m is fixed number for all rounds) straight line segments from anywhere on the circumference of the circle to anywhere else on the circumference of the circle. (I suggest an m of 4 to 6 for beginning players.) The defense player's line segments may cross each other (but don't have to cross).

(I suggest that the bigger m is, the bigger the circle is drawn.)

After the defense player has drawn the m line segments, then it is the offense player's turn to make his/her moves for the round. Clarification: After the defense player draws her/his m line segments, he/she does not move any more during the round.
(So, in a round, first the defense player draws all his/her line segments, then the offense player draws all his/her line-segment-- see below.)

On a move the offense player draws a straight line-segment from an intersection to another intersection*.
*An intersection is either where any line segment (drawn by the defense player) touches the circle, or is where any pair of previously-drawn line segments (drawn by either player) cross.
On every odd-numbered move (the first move, the third move, the fifth move, etc) the offense player's line segment must not cross any other previously-drawn line segments.
On every even-numbered move the offense player's line segment MUST cross exactly one previously-drawn line segment (crossing no fewer, no more than one segment).

And, oh yeah, neither the defense player's nor the offense player's line segments may coincide (coincide along more than one point) with any other previously-drawn line-segment.

The offense player moves until he/she can't move anywhere, otherwise she/he MUST move.
By the way, the defense player may find possible moves for the offense player if the offense player wrongly claims that he/she can't move any more. (It is advantageous for the defense player if the offense player keeps drawing line segments.)

As the offense player draws line segments, the number of these line segments drawn is tabulated.

After playing all the rounds, the winner of this game is the player who, during all rounds that they were the offense player, drew the FEWEST line-segments all together.

--

Question:

I wonder, is there a pattern of line segments the defense player can draw that will guarantee a larger number of moves by the offense player than with any other pattern of line segments drawn by the defense player?
(By "pattern" I mean, as an example, lines drawn parallel, all lines crossing at a center point, the lines forming the perimeter of an m- gon, etc.)

Thanks,
Leroy Quet

PS: After I post this game to my blog, the list of 66 or so games I posted in September will be hidden. Just click on the triangle next to the September link to get that list of games back.

Connectin'-It Game

2008-09-29T17:03:00.001-06:00

This is a game for 2 or more players.
Start with an n-by-n grid taken from graph paper. (I suggest an n of about 10 to 20 for beginners.)

On a move a player can do one of two things:

He/she can draw a dot at any intersection of the grid that doesn't already have a dot and isn't on a line-segment.

Or she/he can draw a straight line-segment that connects any two dots, provided that the line segment does not pass through any other line segment and doesn't pass over any intermediate dots.

A player gets a point every time she/he connects two dots with a line-segment.

The first player to get a pre-determined score wins.
(I suggest a winning score between n^2/4 and n^2/2, if there are two players.)

Thanks,
Leroy Quet

Wind Around Solitaire

2008-09-29T17:02:00.002-06:00

Here is a 1-player game played on an n-by-n grid, where n is odd. (I suggest an n of about 13 to 21 for beginners.)

The player places the numbers 1,2,3,...n^2 IN ORDER into the grid.

1 goes in the center square.

Each number (2k+1) must go either below, above, left of, or right of the square with (2k-1) in it.

Each number (2k) can go in any square that is immediately adjacent to a square with an integer already in it.

Numbers can only be placed in empty squares.

The player's score is the number of times the path of odd integers goes completely around the center square clockwise before the player is unable to place any more numbers in the grid.

Thanks,
Leroy Quet

Slice Through The Boundaries

2008-09-22T15:47:00.000-06:00

This is a game played by any number of people.
It is played on an n-by-n section of grid taken from graph paper. (I
suggest an n of about 12 for beginners, if there are only 2 players.)
Players take turns who is the offense player. (The other players play
defense on a round.) A round is played for each player playing the
game. An empty n-by-n grid is used each round (with the same n as in
the other rounds).
Players take turns each filling in the empty squares of the grid, one
empty square filled in each move by each player.
If there are m players, then each player fills in floor(n^2/(2m))
squares. (That is a total of m*floor(n^2/(2m)) squares filled in all
together.)
Then the offense player draws a straight line (with a straight-edge)
from any side of the n-by-n grid to any other side.
The line must not be perfectly vertical or perfectly horizontal.
The offense player gets a point for every boundary between a filled-in
square and an empty square that the line passes through.
Highest score wins.
Example:
Filled-in square = *. Empty square = o.
n = 6. (View with fixed-width font.)
\ 1 2 3 4 5 6
A o * * * o *
B o * * o o o
C * o * o o *
D * * o o o o
E o o * * o o
F * o * * * *
Let us say that the line goes from just below the upper-left corner of
the grid to just left of the lower right corner. (The line meets the
perimeter of the grid less than one square's length from each of these
corners.)
This is kind of hard to depict here, because the squares of the grid
are literally squares, while they are not in my diagram; but hopefully
it is clear anyway.
The line first crosses the boundary between 1B and 2B. Then it crosses
the boundary between 2B and 2C. Then it crosses the boundary 2C to
3C. Then 3C-3D. Then 4D-4E. Then 4E-5E. And finally, 5E-5F.
Note: Technically we are concerned about the number of times the line
crosses from a filled-in square into an unfilled-in square, or vice
versa. So if a line crosses from a square to a diagonally adjacent
square via the vertex that joins them, then what we are concerned
about is the two squares' status. In other words, the vertex is
considered the "boundary" in that case.
Thanks,
Leroy Quet

Lines-By-Lines

2008-09-22T15:46:00.001-06:00

This is a game for 2 players.
Two rounds are played. In each round one player is offense while the
other is defense.
Players switch who is offense and who is defense for the 2nd round.
Start a round with an n-by-n grid.
Both rounds are played on the same sized grid drawn carefully on
paper.
(It is preferable to use graph paper.)
The offense player moves first in a round.
The first player to move draws a straight line-segment (with a
straight-edge, preferably) from any vertex of the grid to any other,
provided that no intermediate vertexes are crossed. (The only vertexes
to coincide with the line-segment are at the segment's end-points.)
Players take turns each drawing a straight line-segment (from the
vertex where the other player last drew a line-segment to) to another
line-segment. So all the line-segments together form a continuous
path.
Line-segments must not cross or coincide with each other.
Line-segments must not coincide with any vertexes of the grid, with
the exception of at the line-segments' end-points.
Line segments must not travel horizontally or vertically.
Play continues until it is not possible to draw a line-segment to any
other vertex, given the rules.
The offense player gets a point for every square of the grid the path
of line-segments passes through.
(A player gets at most one point for each square, no matter how many
line-segments pass through that square.)
Players switch who is offense, then play another round.
Highest score wins.
Note: It may be more interesting if it is required that the first
player to move draws his/her line-segment from the center of the grid.
Thanks,
Leroy Quet

Triangle Grid Of Coprimality

2008-09-22T15:44:00.000-06:00

Here is a game for 2 players.
The game is played on a "triangle"-shaped grid. 1 square on the top
row; 2 squares on the 2nd row; 3 squares on the 3rd row; etc; n
squares on the nth and bottom row.
(I suggest an n of about 12 for beginners.)

So, we have this:
(Each @ represents a square.)
@
@ @
@ @ @
@ @ @ @
@ @ @ @ @
etc
The players take turns filling in squares of the triangle, one empty
square filled in per move.
Each player fills in floor(n^2 /8) squares. (So, 2*floor(n^2 /8)
squares are filled in by both players together.)
Player 1 gets a point for every COLUMN of the triangle where the
number of filled in squares (filled in by either player) is coprime to
the number of total squares in the column.
Player 2 gets a point for every ROW of the triangle where the number
of filled in squares is coprime to the number of total squares in the
row.
(And for the purposes of this game, 0 is considered to be coprime only
to 1.)
For example, let's say we have a triangle filled like this at the
game's end:
(n= 6. @ = empty square. X = filled square.)
X
@ X
X X @
@ @ X @
X X @ @ @
@ @ X @ @ @
Player 1: column 1 (from left), 3 is not coprime to 6 (no point);
column 2, 3 is coprime to 5 (1 point); column 3, 2 is not coprime 4
(no point); column 4, 0 is not coprime to 3 (no point); column 5, 0 is
not coprime to 2 (no point); column 6, 0 is coprime to 1 (1 point).
Player 2: row 1 (from top), 1 is coprime to 1 (1 point); row 2, 1 is
coprime to 2 (1 point); row 3, 2 is coprime to 3 (1 point); row 4, 1
is coprime to 4 (1 point); row 5, 2 is coprime to 5 (1 point); row 6,
1 is coprime to 6 (1 point).
So, player 1 gets 2 points, while player 2 gets 6 points. Player 2
wins.
Is there a strategy that guarantees a win for one player?
Thanks,
Leroy Quet

Line-Segments In Circle

2008-09-22T15:42:00.000-06:00

Here is a game of mine that is NOT based on an n-by-n grid...
The game is for 2 players.
Draw a circle on a piece of paper. (Preferably the circle is drawn
with a compass, but this is not necessary. The circle should be drawn
carefully, however.)
Put a dot at the circle's center.
Players take turns drawing straight line-segments (preferably with a
straight-edge) within the circle as follows:
*A segment can go from the dot at the circle's center to the edge of
the circle.
*A segment can go from {an intersection of another segment and the
circle} to {another intersection of a segment and the circle}.
*A segment can pass through at most ONE line-segment that has already
been drawn.
*A segment can not coincide with a previously drawn segment. (ie. Two
segments can coincide at at most one point.)
*A segment cannot be drawn from the center to the circle that is 180
degrees away from the segment last drawn by the other player, if the
last segment was drawn from the center to the circle. (This prevents a
player from simply matching the other player's moves, and so
automatically winning.)
The last player able to move is the winner.
Is there a strategy that makes this game easy to win?
Thanks,
Leroy Quet

Turn It Around -- Multiply/Add Grid

2008-09-22T15:39:00.000-06:00

Here is another game played on an n-by-n grid drawn on paper,
where n is even.
(Actually, it would be MUCH easier to play this game on a computer.)
2 players.
Player 1 starts the game by placing a 1 in any square of the grid.
The players take turns placing numbers in the empty squares of the
grid, one number in one square per move.
Integers can only be placed in any square that is adjacent (in the
direction of up, right, down, left) to a square that is already filled
in with a number. The integer placed in a square must be exactly 1
more than the number in any filled-in square that is adjacent to the
square being filled in on the move.
Play continues until all squares are filled in.
Scoring:
For player 1, imagine another grid filled in the same way as the game
grid. Rotate the the second grid 180 degrees and place on top of the
first grid. Now multiply each number in the second grid by the number
immediately below it, getting n^2 total products. Player 1 gets the
sum of these products as the score. (So, the score is relatively
large, as far as my games go.)
For player 2, do the same thing, but rotate the top grid by only 90
degrees. (It doesn't matter if you rotate clockwise or
counterclockwise.)
Here is a small example:
n = 3.
Finished grid looks like this:
1 2 3
4 3 6
5 4 5
Rotated 180 degrees, multiply and add:
Player 1's score =
1*5 + 2*4 + 3*5 +
4*6 + 3*3 + 6*4 +
5*3 + 4*2 + 5*1 =
113.
Rotated 90 degrees, multiply and add:
Player 2's score =
1*5 + 2*4 + 3*1 +
4*4 + 3*3 + 6*2 +
5*5 + 4*6 + 5*3 =
73.
Player 1 wins.

Thanks,
Leroy Quet

Unique Count In Rows/Columns

2008-09-22T15:37:00.000-06:00

This game is for 2 players.
Start by drawing an n-by-n grid on paper. (I suggest an n of about 12
for beginners.)
Players take turns filling in the grid's squares, one empty square
being filled in on each move.
Each player fills in a total of floor(n^2 /4) squares.
(So a total of 2*floor(n^2 /4) squares are filled in all together.)
Scoring is as follows:
Player 1 gets a point for every ROW with a unique number of filled-in
squares in it. In other words, every row that counts does not have a
number of filled-in squares that any other row also has.
Player 2 gets a point for every COLUMN with a unique number of filled-
in squares in it.

Variation:
Players on their moves each draw cards from a regular card deck (no
jokers).
The cards are not placed back, unless n is big enough that the card
would run out -- in which case just reshuffle and reuse the deck as
needed.
The player moving then writes down the number of the card in the empty
square, instead of simply filling the square in.
(Ace = 1, jack = 11, queen = 12, king = 13.)
Then, as before, each player fills in floor(n^2/4) squares. Scoring is
as follows:
Player 1 gets a point for every ROW with a unique SUM of the values of
the filled-in squares in it. In other words, every row that counts
does not have a SUM of the values of filled-in squares that any other
row also has.
Player 2 gets a point for every COLUMN with a unique sum of the values
of the filled-in squares in it.
Thanks,
Leroy Quet

Up And Up, Grid Solitaire

2008-09-22T15:34:00.000-06:00

This game is played solitaire. (If more than one player wants to play,
then each player plays this game with the same sized grid, and players
compare final scores.)
A player starts with an n-by-n grid drawn on paper. (I suggest an n of
about 8 for beginners.)
The player starts the game by placing a "1" in any square of the grid.
The player, on move number m (m = positive integer), places the number
m in an EMPTY square. Square number m must be either left of, right
of, above, or below square number (m-1), for all m >= 2.
On move 2, the player places the 2 one square from the 1. On move 3,
the player places the 3 two squares from the 2. On move 4, the player
places the 4 three squares from the 3, etc.
Now, k is a variable that increases by 1 on each move, on occasion
being set back to 1 (see below).
A player MUST place the number m a total of k moves from the (m-1), if
the (m-1) was (k-1) moves from the (m-2), UNLESS the player cannot do
so (either because there are no empty squares k squares from the
(m-1), or k squares would be off the grid in all directions).
If a player cannot put an m, for whatever reason, exactly k squares
from square number (m-1), then the count starts over at k=1, and the
player places a number m in an empty square ONE square above, right
of, left of, or below square (m-1).
Then k then becomes 2. And on the next move, the player fills in the
square two squares from the last square, etc.
Play continues until the player cannot move any more, even if k is set
back to 1.
The player's score is the number of squares filled = the last number
written in a square.
Sample game: 6-by-6 grid.
23 * * * * 6
9 15 14 8 10 7
* 3 12 2 11 *
18 16 * 1 17 19
22 * * 21 * 20
24 4 13 * * 5
Score = 24.
Math question: What is the highest possible score for a given n-by-n
grid? Is there an interesting sequence as n = 1,2,3,4, etc... Or is
there an easy pattern to the highest possible scores?
Thanks,
Leroy Quet

Crossing The Rings

2008-09-22T15:32:00.000-06:00

Unlike most, but not all, of my earlier games, this game is NOT played
on an n-by-n grid.
Instead, carefully draw (preferably with a compass or a computer) n
equally-spaced concentric circles, where n is about 5 or more for
beginners (and when just 2 players are playing), a much larger n for
advanced players or if there are more players. (The concentric circles
should end up looking something like a target.)
(I suggest you photocopy the concentric circles, so as to make playing
multiple rounds much easier.)
Players (any number >=2) take turns being the "divider".
The divider, on his/her round, subdivides each ring between each pair
of consecutive circles into anywhere from m to 2^m sections, where m
is the order of a ring from the center of the "target". (The range of
allowable numbers of sections per ring can be modified, if players
chose.)
The line-segments dividing the rings should be perfectly straight, and
should meet the consecutive circles they connect at a right angle to
the tangent of the circles at the points where the circles and the
line-segments meet.
But it is up to the divider as to where along the rings the dividing
line-segments go exactly.
Next, the players who are not the divider take turns each drawing a
straight line-segment (with a straight-edge) from any intersection
where a dividing line-segment and a circle meet to any other such
intersection.
The player who is moving then fills in any sections (section = the
sections of the rings that are subdivided by the dividing line-
segments) that his/her line passes through using a colored pencil of a
color different from the colors of the other players' pencils.
The players' lines may intersect each other, and may pass through
already-filled-in sections. But a player can only fill in sections
that are not yet filled in.
Lastly, the divider then draws a line segment from any intersection to
any other, and fills in the sections his/her line passes through. (As
before, the players' lines may cross, but only sections not filled in
before may be filled in.)
Each player gets a point for each section of his/her color.
Rounds are played -- each round with the same number of concentric
circles -- so that each player plays the divider the same number of
times. Players also switch the order they play on each round, so that
each player plays in any given order the same number of times as every
other player does.
Scores are added from the rounds, and the player with the greatest
number of points wins, of course.
Thanks,
Leroy Quet

PS:
Clarification: Either, players should be banned from drawing their
lines to coincide with dividing
line segments, or if a game line and a dividing line segment coincide
then the player drawing the game line should not be able to fill in
either segment on the two sides of the dividing line-segment.
(The players can chose which game-rule they chose.)

Slice And Fill

2008-09-22T15:29:00.000-06:00

Another game from Leroy Quet Inc.:
Start with an n-by-n grid either on graph paper or drawn carefully. I
suggest a relatively small n as far as my games usually go, about 4 to
11.
This game is for two players, each player with a pen/pencil of a
different color than his/her opponent's.
On each move, a player must first take one action (see below), and may
take a second action as well.
First, on her/his move, a player MUST fill an empty section of the
grid entirely, where the section is defined by* the lines of the grid
and/or by straight line-segments drawn by players (see below -- a
section may be a square, or it may be a polygon which is a subset of a
square).
*[By "where the section is defined by...", I mean "where the section is
BORDERED by" the lines of the grid
and/or by straight line-segments drawn by players.
There are no internal line-segments within any given "section" (at
least until lines are added later in the game, crossing the section
and subdividing the section into plural sections). ]
Unless this is the first move by a player or in case of some
other circumstances (see below), then the player must fill in a
section that is adjacent to the last section filled in by the same
player. (By "adjacent", the new section must be a previously unfilled
section that touches the player's last filled-in section along a line,
and not just touching at a point.)
Second, a player MAY on his/her move, after filling in a section,
connect any two vertices of the grid with a straight line-segment
(drawn carefully) (The vertices at the endpoints of any line-segment
must be contained within the n-by-n grid, possibly being on the grid's
border. Line segments may be diagonal, of course.) The line segment
may not cross another line segment previously drawn by either player
as part of the game. But line segments can cross grid-lines and cross
previously filled-in sections.
If a player fills in a section that is adjacent (touching along a
line, not just at a point) to a section previously filled in by the
player's opponent, a "point" is given to the player. (I put "point" in
quotes, because the goal of the game is to get as FEW points as
possible.)
Also, if a player has just filled in a section that is adjacent to a
section previously filled in by the player's opponent, then the player
may fill in any empty section of the grid on her/his next move,
starting a new path of adjoining sections.
The previous paragraph tells one situation where a player can fill any
section of the grid, not necessarily a section next to the section
previously filled in by that player. The other situation is when the
player cannot move because all adjacent sections to the player's last
filled-in section are already filled in.
Play continues until all sections are filled in.
The winner has the FEWEST number of points.
Note: Officially, the line-segments and grids act as if they were
drawn perfectly -- the line-segments pass through the appropriate grid-
vertices, given the slope of the lines.
Any strategies for this game? Any way to ensure a win for one player
or the other?
(If so, I need to fix the rules.)

Thanks,
Leroy Quet

(Convex) Hull Lot Of Fun

2008-09-22T15:23:00.000-06:00

Start by carefully drawing a square grid that has an odd number of
lines -- and an even number of squares -- on each side.
I suggest about 9 or more lines -- 8 or more squares -- on each side,
for beginners.
I suggest bigger grids with more lines for advanced players.
The game begins with the players (two in number) taking turns drawing
dots, one dot per move, each dot at an empty (no dot there yet)
intersection of grid-lines.
Each player draws some fixed number of dots. I suggest that, if there
are m^2 intersections in the grid (where m is the number of lines on a
side of the grid), then each player draws m^2/6 (approximately) dots,
for m^2/3 total number of dots.
Player 1 moves first, followed by player 2, then player 1, then player
2,....
On the first move by player 1, the dot CANNOT go in the center
intersection of the grid.
After the dots are drawn, either player draws line-segments to connect
the dots in the boundary of the convex hull of the dots. (See below
for convex hull definition.) -- The convex hull boundary connects the
same dots that it would connect if the grid happened to have been
perfectly drawn. (So, this game might be a good teaching tool for
learning slopes.)
(By the way: If the player not drawing the line-segments thinks that
the "convex hull's boundary" being drawn is not exactly the true
convex hull boundary, because it is connecting the wrong dots, then
the "convex hull's boundary" can be challenged for accuracy.)
After the first convex hull boundary is drawn, the boundary of the
convex hull of all the dots inside (and not on the edge of) the first
convex hull boundary is drawn.
Then the next convex hull boundary within the previous boundary is
drawn...
This continues until in the center of the innermost convex hull there
are zero dots, 1 dot, or several dots in a line.
If there are an odd number of dots in the center, then player 1 wins.
If there are an even number of dots in the center, then player 2 wins.
I suggest that players play an even number of rounds, and switch who
is player 1 and who is player 2. Then who wins the most number of
rounds is the winner. (This suggestion is in case there is a bias in
this game towards either player 1 or player 2.)
What is a good strategy for this game?
Convex hull: A convex hull of a set of points in a plane is the
smallest CONVEX polygon that contains all the points. (Convex has
pretty much the intuitive meaning here: There are no concave sections
along the polygon's boundary. Also, ANY point within the polygon can
be connected with ANY other point within the polygon by a straight
line-segment without the line-segment ever leaving the polygon.)
Intuitively: (Following plagiarized from Wikipedia.) Imagine the dots
of the game being pins sticking out of a board. Imagine an elastic
band that stretches around all the pins. Releasing the band, it
contracts around the outermost pins to form the boundary of the convex
hull.
By the way, I know that many newsreaders block posts made from Google
(like this post) because of all the spam coming from Google.
The question is, has anyone seen this post? Or is Google blocked by
about everyone now?
Thanks,
Leroy Quet

PS:
I feel I should mention in words why I have the rule that the player 1
cannot put a dot at the center intersection of an odd-by-odd grid on
the first move.
Say player 1 places his first dot at the center intersection. Then
player 2 puts her dot anywhere else. Then player 1 needs only to
continue to match player 2's moves so that the finished grid has
rotational symmetry.
Then, either there will be one dot inside the center convex hull at
game's end, or there will be a line of an odd number of dots. Player 1
automatically wins.
It would be advised for player 2 to try to continually keep the grid
from getting symmetric about any point (not necessarily the very
center dot) so that player 1 cannot just reflect player 2's moves
about the point.
If we have an even number of lines on each side of the grid, then
player 2 simply matches player 1's moves so that the grid has
rotational symmetry. Player 2 wins automatically.
I wonder if there are any necessary rule changes that are needed to
prevent easy-wins for one side or the other. (By easy-win, I mean some
strategic move that would ensure a win for a given player.)
Thanks,
Leroy Quet

Another Criss-Cross Grid Game

2008-09-22T15:17:00.000-06:00

Here is another one of my games. It isn't as fun, probably, as some of
the other games I have posted. But maybe someone will find it
enjoyable anyway.
Start with an n-by-n grid (n-by-n lines, (n-1)-by-(n-1) row/columns)
on graph-paper. (n should be at least 5, maybe in the range of 10 or
more.)
There are two players who switch roles after each round, an offensive
player and a defensive player.
Say the grid is n lines wide((n-1) columns) and n lines high ((n-1)
rows). The defensive player starts the round by writing the integers 1
through (2n) in any order to the left of the left-most vertical line
(each integer lined up with a different horizontal line of the grid)
and above the top-most horizontal line (each integer lined up with a
different vertical line of the grid).
Whether a particular integer is written either along the left side of
the grid or along the top of the grid is up to the defensive player.
Example: 6 lines -by- 6 lines:
. 2 6 11 7 1 8
9 -------------
4 | + + + + +
10| + + + + +
12| + + + + +
3 | + + + + +
5 | + + + + +
(Note: In case ascii art doesn't look correct, the pluses are the
intersections of the grid, and are supposed to each be directly below
an integer of the top row of numbers and directly to the right of the
left column of numbers.)
The two players each take turns drawing a straight line-segment (with
a straightedge) from the intersection of the grid last drawn-to by the
opposing player to an intersection determined by the order of the move
within the round.
If the move is move m -- the defensive player moves on even-numbered
moves, and the offensive player moves on odd-numbered moves -- then
the player can move to any intersection in the same row/column lined
up with the m along the grid's edges. In other words, if the value m
is written along the top of the grid, then the player on the mth move
can move to any one of the n intersections in the same COLUMN as the
value m. And if the value m is written along the left side of the
grid, then the player on the mth move can move to any one of the n
intersections in the same ROW as the value m.
Players cannot draw line-segments along already drawn line-segments.
(ie Line-segments can only intersect at most at one point.)
Line-segments cannot cross intersections that are already the
endpoints of other line-segments.
After 2n total moves (n moves for each player), the round is over.
The offensive player gets a point for every time a line-segment (drawn
by either player) crosses another segment.
If k line-segments intersect at a common point, the offensive player
gets k(k-1)/2 points for that intersection.
This is the same as counting the number of line-segments intersected
by a line-segment AS the line-segment is being drawn. (If your line
segment is crossing an intersection with (k-1) line-segments already
intersecting there, then add (k-1) to the offensive player's score. Or
wait until after the game is over to enumerate the crossings, and give
the offensive player k(k-1)/2 points for those k line-segments
intersecting at a common point.)
So, the defensive player moves so as to try to keep the lines from
crossing. The offensive player moves to try to get as many crossings
as possible.
After a round is complete the players switch who is the offensive
player and who is the defensive player. The highest score wins, after
playing an even number of rounds, of course.
What is a good strategy for this game?
Thanks,
Leroy Quet

Quite A Stretch -- Up/Down Card Game

2008-09-22T15:09:00.000-06:00

Here is a card game for two players. (Although it could be easily
modified for more players.)
This game seems slightly familiar to me. Are major aspects of it taken
from pre-existing card games?
All that matters in this game as far as the cards are concerned is
each card's numerical value (with Ace = 1, Jack = 11, Queen = 12, King
= 13).
Start by shuffling a deck of playing cards. (No jokers.)
Divide up the cards evenly between players. A player is not allowed to
see his/her opponent's cards until the cards are played. (The cards
belonging to each player that have not yet been played I will call a
player's "hidden hand".)
All cards once played in this game are placed face-up.
Each player starts their pile of cards by placing any one card they
choose (face-up, of course) down between the players. (So we have two
piles, one for each player.)
On each round (of two moves each) players take turns being the
offensive player and the defensive player.
The offensive player puts a card down (face-up) next to his pile. Say
that the top card (ie, the last card played by that player in the
previous round) in the offensive player's pile has a numerical value n
and the card he/she just put down next to the pile has a value m. And
say the card on top of the defensive player's pile has a value k. Then
the defensive player must, if he/she can, place a card (any card he/
she chooses from his/her hidden hand) down on top of his/her pile that
differs from k by less than or equal to the absolute value of (n-m)
AND is in the same numerical direction from k as m is from n. So, in
other words, if the card the defensive player plays has a value j,
then (m-n) has the same sign (+, -, or is zero) as (j-k).
And |m-n| is >= |j-k|. (|j-k|, the absolute value of the difference between the
card the defensive player last played and the card the defensive
player is now playing, must be any value from 0 to |m-n| {ie, from 0
to the absolute difference between the card that the offensive player
played in the previous round and the card the offensive player has
currently played}.)
(See example.)
If the defensive player cannot move, then he/she skips his defensive
move, and therefore does not remove a card from his/her hidden hand in
that round.
The players then move the cards that are next to their piles onto the
top of each pile (face-up).
Then, for the next round, the players switch who is offense and who is
defense, and the formerly defensive player then plays offense.
(So players play cards like this {if both players can move}: {player
1, player 2}, {player 2, player 1}, {player 1, player 2}, {player 2,
player 1}, etc.)
(Even if the defensive player did not play a card in the previous
round, the defensive player then immediately becomes the offensive
player of the next round and plays a card then anyway, whatever the
outcome of the previous round.)
The first player to run out of cards in his/her hidden hand wins.
Here is the beginning of a sample game:
Start: Player 1 puts down a 5, and player 2 puts down a 6.
Round 1:
Player 1 puts down a 10. (So player 2 must put down a 6,7,8,9,10, or
11.)
Player 2 puts down a 7.
Round 2:
Player 2 is now offense, and he puts down a 3. (So player 1 must put
down a 6,7,8,9, or 10. -- Since the card put down in round 1 by player
2 was a 7, and last card put down by player 1 was a 10.)
Player 1 places down an 8.
Round 3:
Player 1 then places down another 8. (So player 2 must put down a 3.)
Player 2 does not have a 3, so player 2 skips defensive move.
Round 4:
Player 2 places down a 7. (So player 1 must put down a 8,9,10,11, or
12. -- Note: Last card played by player 2 was a 3, from round 2.)
Player 1 places down a 12.
ETC.
What would be a good strategy for this game?
Thanks,
Leroy Quet

Horizontal/Vertical (Sometimes Diagonal) Grid Game

2008-09-22T15:07:00.000-06:00

Start with an n-by-n grid drawn on paper. (I suggest an n of at least
8, if not much larger. But not too large if you don't have a long time
to play.)
The players (2 in number) take turns filling in the squares of the
grid, one square per move.
Player 1 fills in the first square anywhere in the grid.
Play continues like this:
Player 1 fills in any EMPTY square that is either left of or right of
the last square filled in by player 2.
Player 2 fills in any EMPTY square that is either above or below the
last square filled in by player 1.
Either player may fill in an empty square that is diagonally touching
the last square filled in by the other player IF BOTH players agree
that such a move is acceptable at that time.
The players alternatingly fill in a string of squares this way until
one player cannot move. (If a player can move, the player must move.)
Then the player that could not move fills in any empty square with his/
her initials or unique symbol (chosen before play). (The other player
then moves from that position as before in the next move.)
(Note: by "empty" square, I mean a square that neither has been filled
in nor has any initials or symbols in it.)
Play continues until all squares (including isolated single squares)
are filled in or have symbols/initials in them.
The player with the MOST symbols/initials is the winner.
So it is good to NOT be able to move as much as possible during play.
Note: In the first move of the game, player 1 does NOT write his/her
symbol/initials into the first square. Only after a player cannot move
does that player write their symbol/initials in a square.
Clarification: If a player is in a situation where he cannot move
either (up/down)(left/right), but he/she can move diagonally, then
that player must move diagonally IF the other player agrees to this
move. Most often the other player will agree, since this denies the
first player a point. But sometimes for strategic reasons the other
player may deny the first player the right to move diagonally.
What would be a good strategy for this game?
Thanks,
Leroy Quet

Move By The Numbers

2008-09-21T07:55:00.000-06:00

This game is for two players, and it involves (as do many of my games)
an n-by-n grid drawn on paper.
First, each player makes a list of n positive integers (where n is
the number of squares along a side of the grid), each integer in the
list being <= n-1. (The order of the integers is significant.) Each
player makes his/her list without knowlege of the other player's list.
Players then write their numbers, in order, along the edges of the
grid.
Player 1 writes her/his integers above the top row of the grid,
exactly
one integer above each square. Player 2 writes his/her integers to the
left of the left-most column of the grid, exactly one integer to the
left of each square.
One of the players plays offense, the other defense. After the round
is
complete, the players switch who is offense and who is defense, using
the same numbers in the same order, but using a new unmarked grid.
To start, player 1 places a "1" in any of the grid's squares.
Players take turns placing integers in the grid. Player 1 places
1,3,5,7,..the odd positive integers, in order, in the grid. Player 2
places 2,4,6,8,... the even positive integers, in order, in the grid.
A player, on move k of the game, places the integer k in any EMPTY
grid
square. He/she places the k directly to the right, to the left, above,
or below the square with (k-1) in it. ((k-1) is in the last number put
in the grid by the other player.)
Let the integer written above the column the integer (k-1) is written
in
be c. Let the integer written to the left of the row (k-1) is written
in
be r.
The player placing the integer k in a square must place that integer
EITHER c or r squares (in any of the 4 main directions) from the
square
the (k-1) is written in.
(The side of the grid that an integer is written next to does not have
anything to do with what direction the k-square is from the (k-1)-
square.)
The players act as if the top and bottom of the grid are connected,
and
act as if the left and right sides of the grid are connected.
(Toroidal
topology.)
So if a move, say, is off the grid to the right, the players acts as
if
the row forms a circle, and continues counting from the left side,
counting to the right.
A move to the left off the grid continues from the right side on the
same
row, continuing to the left. A move upward off the grid continues from
the
bottom of the same column, continuing upward. And a move downward off
the
grid continues from the top of the same column, continuing downward.
And remember. All moves must end up on empty squares.
If a player can move, she/he must.
The round is over when the players can't move anymore.
The offensive player gets a point for every square that has a number
in it.
(Ie the offensive player gets a score equal to the largest number in
any
square of the grid.)
The players switch who is defense and who is offense, as I said above,
with
the same numbers along the edges of the grid, but with a new unmarked
grid.
Highest score wins.

Thanks,
Leroy Quet

Claim The Grid's Squares

2008-09-21T07:51:00.000-06:00

Game for 2 players.
Draw the grid large enough so that each square can contain
the game-piece (such as a coin) of each player.
Play starts with each player marking any one of the squares,
a different square for each player, with the player's symbol
or filling in a square, using colored pencils or pens, with
the player's color.
Players each then place their game-piece on their starting
square.
The game consists of "turns" where one player moves and then
the other. Who moves first in each turn alternates. So the
moves are done like this:
(player 1, player 2) (player 2, player 1) (player 1, player 2) (player
2, player 1), etc.
Before either player moves in a turn, the player who moves
second in the turn calls out how many spaces both players
will move in the turn. The number of spaces called out
is an integer from 1 to (n-1).
Players, in the appropriate order, then move their game-
pieces either up, down, left, or right the same number of
squares on the grid. (Both players must move the same number
of positions in a turn, but each can move in their own
direction.)
If a player cannot move because the number called is too big,
and the move would take the player off the grid, then that
player simply does not move on that turn.
When a player is the first to land his/her game-piece on a
square, the player then marks that square with his/her symbol,
or fills the square in with her/his color.
Players can always land on squares that are marked (by either
player), but they can only claim empty squares for their own.
Players can't move onto squares where their opponent's game-
piece is located.
Play continues until every square is filled in, or until a
predetermined number of turns have passed.
The winner of the game has the most number of squares marked
with their symbol or with their color.

Clarification: a player must move if he/she is able to move, whether she/he wants to or not.

Thanks,
Leroy Quet