Consider a 2x2 board in which each square can be black or white. Obviously there are 16 different 2x2 patterns, counting reflections and rotations.
A 5x5 board has 16 different 2x2 subsquares. Can the squares of a 5x5 board be colored so that each 2x2 subsquare has a different pattern?

(In reply to Solution by Penny)

I set this up in a spreadsheet with all 16 subboards (as 2x2 tiles) and a 5x5 grid.

Using Penny's arrangement I cut and pasted each tile to the 5x5 grid and I agree that Penny has a solution.

In this arrangement there are 14B and 11W.

The 16 tiles/subboards have 32B and 32W.  They can grouped as pairs of inverses, eg 4B with 4W.

Because of this property the B's and W's can be interchanged giving  a solution of 14W and 11W. [Then we have the obvious rotations and reflections of both of these arrays].

Posted by brianjn on 2007-04-20 20:46:14