67th NMO Selection Tests for JBMO

Question

All the 16 squares of a 4 4 array are white. Define a *move* by selecting a rectangle 1 3 or 3 1 and switching the colors of each of its squares from white to black or from black to white. Is it possible that all squares turn black after a sequence of moves?

Accepted Answer

1	2	3	1
2	3	1	2
3	1	2	3
1	2	3	1

to observe that a move will change colors in one square of each number. As initially there are six squares labeled 1, an even number of moves is required to turn black all these squares. On the other hand, five squares were labeled with 2 at the start, requiring an odd number of moves to turn all black, hence the answer is negative.