Tuesday, March 17, 2009

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.

Leroy Quet

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