Saturday, September 20, 2008

Guessing # Of Uncoprime Walls

Another one of my less-than-famous math games played using a square
(Maybe this game can, like several other of my games,
help teach students about coprimality.)
Game is for 2 or more players.
Start with an n-by-n grid drawn lightly on paper.
Players take turns placing the integers 1 through n^2 in the
grid's squares, one integer per square. (So, if there are 2 players,
player 1 places the odd integers in the grid, player 2 places
the even integers in the grid.)
Each integer is written in any blank square that is immediately
adjacent (in the directions of either up, down, left of, or
right of) to a square with an integer already in it.
(Player 1 may put the 1 in any of the grid's squares.)
Before the players start to fill in the grid, however, they
each secretly guess how many "walls" will, at game's end,
separate adjacent integers which are not coprime, and then
they write this guess down (not showing their guess to any
other player until the game is over).
(A wall is a vertical or horizontal line-segment of the grid,
of one grid-square in length, separating two adjacent grid-squares.)
After the grid is filled, every wall within the grid separating
an adjacent pair of non-coprime integers is darkened in.
The player whose guess is closest to the actual number of walls
separating non-coprime adjacent integers is the winner.
(Ties are possible.)
(Players may use, as part of their strategy, especially with larger
sized grids, bluffing:
For instance, making it seem they picked a bigger number for the
number of uncoprime pairs than they actually picked, for example,
by at the game's beginning placing many integers next to other
integers they are not coprime to.)
Play m (where m is the number of players) rounds,
then add up the differences between any player's guesses and the actual
number of walls in each round.
The winner is the player whose sum of differences is the *lowest*.
Leroy Quet

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