Some unit squares on an infinite sheet of squared paper are colored green such that:

Every 2x3 rectangle contains precisely two green squares, and:
Every 3x2 rectangle contains exactly two green squares.

How many green squares are there in a 9x11 rectangle?

*** Based on a Russian Math Olympiad problem.

Gut reaction: 1/3 of the squares are green so 1/3*9*11= 33 green squares in a 9x11 rectangle.

Is such an arrangement possible?
I found one way:  Every third diagonal of squares is green.  Each 2x3 or 3x2 rectangle captures either 2 squares from the same diagonal or 1 each of two adjacent diagonals.
In the 9x11 rectangle each row of 9 contains 3 greens. 3*11=33 greens.

Am I missing something?  Those Olympiad problems are usually really hard.

Posted by Jer on 2014-09-23 10:10:57