25th ASU 1991 Problem 8 A minus sign is placed on one square of a 5 x 5 board and plus signs are placed on the remaining squares. A move is to select a 2 x 2, 3 x 3, 4 x 4 or 5 x 5 square and change all the signs in it. Which initial positions allow a series of moves to change all the signs to plus?
Solution
only the central square Solution We take the 5x5 square, the two yellow 3x3 squares, which overlap at the center, and the two blue 2x2 squares. Then every square except the center square is changed an even number of times. So this works if the central square was selected. It is easy to check that any 2x2, 3x3, 4x4 or 5x5 square has an even number of green squares, so if the selected square was green, and we change it an odd number of times, then some other green square must also be changed an odd number of times and hence end up with a minus. So if all the squares end up plus, then the selected square was not green, so it must belong to the central white column. Similarly, it must belong to the central row and hence must be the center square. 25th ASU 1991 © John Scholes [email protected] 14 March 2004 Last corrected/updated 14 Mar 04
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