Question 4
On a $32\times32$ board, a mouse starts at the bottom-left cell facing up, and pieces of cheese are placed in some other cells. The mouse moves forward; whenever it reaches cheese, it eats part of it, turns right, and continues. A set of cheese cells is good if the mouse tastes each piece exactly once and then leaves the board. Prove:
(a) no good set has $888$ cells;
(b) some good set has at least $666$ cells.
(a) no good set has $888$ cells;
(b) some good set has at least $666$ cells.