Sum Of Products, Number Of Numbers Game

Here is a game for 2 players played using an n-by-n grid drawn on paper.

Players take turns. On a turn a player writes any one integer, 1 to n, into any empty square of the grid.

After n^2 total turns, the game is over.

Player 1 gets the sum of the scores for each row. The score for a row is (the number of 1's in the row) * (the number of 2's in the row) * (the number of 3's in the row) *...(the number of m's in the row), where m is the largest integer such that all integers 1 through m occur in that row.

Player 2 gets the sum of the scores for each column. The score for a column is (the number of 1's in the column) * (the number of 2's in the column) * (the number of 3's in the column) *...(the number of m's in the column), where m is the largest integer such that all integers 1 through m occur in that column.

A player gets 1 point for a row/column if there are no 1's in that row/column.

The player with the largest score wins.

And example game:

1 1 2 1 2 5
1 3 2 1 3 5
5 1 1 1 3 1
1 2 1 2 1 2
2 1 1 2 1 1
2 3 5 3 4 1

Player 1 gets:
3*2 + 2*1*2 + 4 + 3*3 + 4*2 + 1*1*2*1*1
=
33 points.

Player 2 gets:
3*2 + 3*1*2 + 3*2 + 3*2*1 + 2*1*2*1 + 3*1
=
31 points.

Player 1 wins.

Thanks,
Leroy Quet

Binary Scramble Game

Here is a game for any plural number of players.

A list of 0's and 1's is written on a piece of paper by the players taking turns, each player appending a 0 or a 1 onto the right side of the list each turn. After a predetermined number of turns (which is a multiple of the number of players), the first part of this game is over.

In the second part of the game, the players take turns rewriting the entire list each turn with one digit the player chooses flipped from 0 to 1 or from 1 to 0.

The new list cannot match any list previously arrived at during the game.

If the lengths of the runs of 0's and 1's form a permutation of the lengths of the runs from any previous list, then the currently moving player gets a point.

(It doesn't matter if a particular run-length, an element in the permutation, was for a run of 0's or for a run of 1's.)

The game continues until either a player first achieves a predetermined score or until no more moves are possible.

The player with the greatest score wins.


Example game (to start):

001110101 (start: 2,3,1,1,1,1)
000110101 (3,2,1,1,1,1 point)
000111101 (3,4,1,1)
000011101 (4,3,1,1 point)
100011101 (1,3,3,1,1)
101011101 (1,1,1,1,3,1,1)
111011101 (3,1,3,1,1 point)
111010101 (3,1,1,1,1,1,1 point)
111010111 (3,1,1,1,3 point)
... etc.

Thanks,
Leroy Quet