APMO 2021 P4 - There exists a good subset consisting of at least $666$ cells
Posted: Wed Jun 09, 2021 5:23 pm
Given a $32\times32$ table, we put a mouse (facing up) at the bottom left cell and a piece of cheese at several other cells. The mouse then starts moving. It moves forward except that when it reaches a piece of cheese, it eats a part of it, turns right, and continues moving forward. We say that a subset of cells containing cheese is good if, during this process, the mouse tastes each piece of cheese exactly once and then falls off the table. Show that:
- No good subset consists of $888$ cells.
- There exists a good subset consisting of at least $666$ cells.