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Disjoint squares

Disjoint squares game. Use mouse wheel or numeric keys to change the square type (if not enough, a .. z starts working. Left mouse click paces the number on the square and right click removes them.

Disjoint squares is a game, created by Prof Terence Tao. This game arose from the study of sums and differences of numbers in a finite set. It has application to the Kakeya problem in combinatorial geometry.

The goal of this game is to place as many numbered squares on the board as possible. There are following restrictions:

  • The number of square types cannot exceed the size of the board. Thus on a 8x8 board, only the numbers from 1 to 8 are allowed.
  • The two squares with the same number cannot share the same row or column. Areas that are forbidden by this rule are marked in red. These areas change when you select another current number.
  • If two squares A and B have the same number, then no other numbered square can share both a row with A and a column with B. Areas that are forbidden by this rule are marked in black.

The score is measured as

,

where N is the number of numbered squares, and n is the size of the board. The highest score ever achieved is log(6)/log(3)=1.63... It can be attained whenever n is a power of 3.

It's known that one cannot exceed 25/13 = 1.923.... I you would succeed, we are highly interested in knowing the solution.