MathDB

4

Part of 2010 Junior Balkan Team Selection Tests - Moldova

Problems(1)

equal sum in each line and in each column, in 5x5 square

Source: 2010 Moldova JBMO TST p4

2/25/2021
In the 2525 squares of a 5×55 \times 5 square, zeros are initially written. in the every minute Ionel chooses two squares with a common side. If they are written in them the numbers aa and bb, then he writes instead the numbers a+1a + 1 and b+1b + 1 or a−1a - 1 and b−1b - 1. Over time he noticed that the sums of the numbers in each line were equal, as well as the sums of the numbers in each column are equal. Prove that this observation was made after an even number of minutes.
combinatorics