Disjoint squares

The object of this game is to place as many numbered squares on the board as possible. There are of course several limitations:

The score is measured as log(N)/log(n), 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.., which one can attain whenever n is a power of 3. On the other hand, it's known that one cannot exceed 25/13 = 1.923....

To place or remove a square, simply click on the board. To change the numbering of the square, press the appropriate numeric key (1-9) or use the choice bar on the right-hand side. Blacked out squares cannot be occupied by any new square; red-colored squares cannot be occupied by squares with the currently selected number.

Press the space bar to clear the board.

On the right I've displayed the number of squares you need to beat the world record of log(6)/log(3); if you ever do beat this, please save the configuration to the Java console using the 'S' key, and send me an e-mail!

This game arose from the study of sums and differences of numbers in a finite set, and has application to the Kakeya problem in combinatorial geometry.

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