Monday, March 21, 2011

Sum Kind Of Game

This is a game for two players.
Start with an n-by-n grid drawn on paper, where n is even.

Players take turns placing numbers into the empty grid squares, one number per move.
Player 1 places into the grid a 1 then a 2 then 3 then... then the value of n^2/2-1 then finally the value of n^2/2.
Player 2, on the other hand, goes the opposite way, placing into the grid the value of n^2/2 then the value of n^2/2-1 then... then 3 then a 2 and finally a 1.

Player 1 gets as a score the number of squares with integers S where each is the sum of the values in two different squares of the same ROW (same row as their sum S).

Player 2 gets as a score the number of squares with integers S where each is the sum of the values in two different squares of the same COLUMN (same column as their sum S).

(The addends can be the same value, but must be in different squares.)

It doesn't matter who wrote down each number as far as scoring is concerned.

Largest score wins.

Sample game: (n=6)

01 02 03 04 05 18
12 17 05 15 11 10
10 14 13 17 03 06
09 14 06 04 09 02
08 16 07 15 08 12
16 18 07 13 11 01

Player 1 gets for each row (top to bottom):
3, 2, 2, 1, 2, 1
11 points.

Player 2 gets for each column (left to right):
2, 2, 1, 1, 3, 2
11 points.

(I may have made a mistake or two figuring these out.)

A tie, if I didn't err.


Is there a bias in favor of one player in the game because strategies are different due to the two orders the numbers are placed into the grid?

Leroy Quet

Saturday, March 12, 2011

Flipping Bits/Squares

This game is for any plural number of players.

You need a blank piece of paper and a pen/pencil.

A predetermined number of rounds are played.

Start each round by making a row of n squares, where n is at least 7 or more if there are 2 players; n is larger if there are more players.
Fill in every other square.

Take turns.

On a turn a player can change any ONE square from filled in to not filled in, or vice versa.
Make a new row of squares to reflect the move.

A move may not lead to a pattern of how the squares are filled that has already existed in the round.
If all patterns achievable by changing one square (flipping one bit) lead to patterns that have already existed in the round, then a player flips two bits. If all patterns achievable by flipping two bits also lead to patterns that already occurred in the round, then flip three bits; etc.
A player MUST move with the fewest number of bits flipped as possible that will lead to a new pattern for the round.

As soon as a player achieves with their move a pattern with exactly one bit/square a different color (either filled or not filled) than the rest of the row, then that player gets {the number of squares from the left side of the row that is this unique square's position} added to his/her score.

The round is then over.
(So, one player scores each round.)

Play a predetermined number of rounds, adding up each player's scores.

The player with the largest grand total wins.

Example round: (n = 7)

(*)( )(*)( )(*)( )(*): start
(*)( )( )( )(*)( )(*): player 1
(*)( )( )( )(*)( )( ): player 2
(*)( )( )(*)(*)( )( ): player 1
(*)(*)( )(*)(*)( )( ): player 2
(*)(*)( )(*)(*)( )(*): player 1
(*)(*)(*)(*)(*)( )(*): player 2
Player 2 gets 6 points, since the blank square is the 6th square from the left.

(I wonder what the funnest n is for any given number of players, especially for 2 players.)

Leroy Quet