On each cell of a $9 \times 9$ grid, there is a chip. At a certain signal, each chip jumps to one of the cells touching diagonally the one it is in. Find the minimum number of empty cells after this move.
We color the columns of the table alternately in black and white. There are then $9 \times 5=45$ black squares, and $9 \times 4=36$ white squares. During the movement, each chip changes the color of its column. Thus, there are more than 36 chips on the 45 black squares after the signal, and therefore at least 9 empty squares. On the other hand, it is easy to find a configuration where there are exactly 9.