Move In Synch, In Opposition

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(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

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

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

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

(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

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