Question

In every cell of a board with sizes 5x5 cells sits a bug. At some moment all bugs crawl into neighbouring (on horizontal or vertical) cells. Will an empty cell remain?

Answer 1

As a total number of cells of chessboard with sizes 5x5 is odd, the number of black and white cells can not be equal. Let’s consider that there are more black cells. Then, there are less bugs, sitting on white cells, than on black. That is why at least one black cell will remain empty, since bugs, sitting on white cells, crawl on black cells.