Thursday, June 10, 2010

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.

Leroy Quet

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