Permudrome -- Grid Game

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

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

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

(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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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