Friday, May 29, 2009

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.

Leroy Quet

Wednesday, May 13, 2009

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.

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.