On a computer screen, there is a $98 \times 98$ chessboard, colored in black and white in the usual way. Using the mouse, you can select any rectangle whose sides are composed of edges of certain squares of the chessboard, and click on it. Then all the squares inside the selected rectangle change color: white squares become black and vice versa. Find the minimum number of such clicks needed to make all the squares on the chessboard the same color.

## Solutions

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In this solution, we will replace 98 with any even number $n$. It is simple to find a way to make the entire chessboard the same color. It suffices to first select every other column, then every other row. This takes $n$ clicks. Let's show that this is in fact the minimum we are looking for. We say that a vertex of one of the squares of the chessboard is bad if it is the vertex of an odd number of black squares and good otherwise. At the very beginning, the only bad vertices are those on the edge of the chessboard, except for the two white corners, so there are exactly $4n-2$ of them. At each step, when we select a rectangle $R$, this can only make the corners of this rectangle good. We can therefore reduce the number of bad vertices by at most 4 at each step. But at the end, this number must be zero: we therefore need at least $n$ steps.

## 2 Thursday afternoon and Friday: Geometry

## 1 Thursday afternoon: David Zmiaikou