32nd Swedish 1992 Problem 2 The squares in a 9 x 9 grid are numbered from 11 to 99, where the first digit is the row and the second the column. Each square is colored black or white. Squares 44 and 49 are black. Every black square shares an edge with at most one other black square, and each white square shares an edge with at most one other white square. What color is square 99?
white Solution Note first that if two adjacent squares in a row are the same color, then the colors alternate as you go down the column containing either of the squares (and the two columns containing them match each other). For suppose 48 and 49 are black, then since each has no other black neighbors, 38 and 39 must both be white. Similarly, 28 and 29 must both be black. Hence 18 and 19 must both be black. Similarly going down the columns. A similar result applies if two adjacent squares in the same column are the same color. Now 44 and 49 are the same color, but there are 4 squares between them, so we must have two adjacent squares in the set 44, 45, 46, 47, 48, 49 which are the same color. It follows that their two columns match. Now suppose that 49 and 99 are the same color. Then the same argument shows that two adjacent squares from 49, 59, 69, 79, 89, 99 must be the same color. It then follows that their rows are of alternating colors. But these rows would intersect the two matching columns. Contradiction. Hence 49 and 99 must be different colors. 32nd Swedish 1992 © John Scholes jscholes@kalva.demon.co.uk 24 February 2004 Last corrected/updated 24 Feb 04