5. On a $4 \times 4$ board, there are 16 grasshoppers, each on its own square. At a certain moment, each grasshopper jumps to an adjacent square: a square up, down, to the right, or to the left, but not diagonally and not off the board.

What is the maximum number of squares that can be empty afterward?
A) 8
B) 9
C) 10
D) 11
E) 12
A5. C) 10 We color the squares of the $4 \times 4$ board in a checkerboard pattern and name the squares from a1 to d4 as shown in the figure. We see that the grasshoppers on white squares all jump to a black square, and the grasshoppers on black squares jump to a white square.

![](https://cdn.mathpix.com/cropped/2024_04_17_2ddd9533af815a0f4acbg-2.jpg?height=246&width=243&top_left_y=197&top_left_x=1495)

The squares adjacent to a3, c1, and d4 are all different, so the three grasshoppers from a3, c1, and d4 end up on three different white squares. Similarly, the three grasshoppers from a2, c4, and d1 end up on three different black squares. After jumping, there are thus at least six squares occupied.

We can ensure that only six squares are occupied after jumping. The grasshoppers from the black squares can each jump to one of the squares a2, c4, and d1, and the grasshoppers from the white squares can each jump to one of the squares a3, c1, and d4.

The smallest possible number of occupied squares is thus 6, and the largest possible number of empty squares is $16-6=10$.