14th ASU 1980 Problem 12 Some unit squares in an infinite sheet of squared paper are colored red so that every 2 x 3 and 3 x 2 rectangle contains exactly two red squares. How many red squares are there in a 9 x 11 rectangle?
33 Solution There cannot be two red squares with a common side. For consider as the diagram shows we can immediately conclude that the squares with a * are not red, but now the bold rectangle has at most 1 red square. Contradiction. Consider a red square. One of the two diagonally adjacent squares marked * must be red. But it is now easy to show that all red squares on that diagonal are red and that the other red squares are those on every third parallel diagonal line. Any 9 x 11 rectangle must have just three such diagonals on a 9 cell border row, and hence just 3 red cells in that border row. But the remaining 9 x 10 rectangle can easily be partitioned into fifteen 3 x 2 rectangles, each with 2 red squares. 14th ASU 1980 © John Scholes jscholes@kalva.demon.co.uk 4 December 2003 Last corrected/updated 4 Dec 03