Question 115
Maena and Théodore play a game. They play on a square grid made of $99 \times 99$ cells. Two cells are considered adjacent if they share a vertex or a side.
Initially, Maéna numbers the cells of the grid from 1 to $99^{2}$, in an arbitrary way. Théodore then places a token on one of the cells of the square, and then allows himself moves of the following form: he may move the token from one cell to another only if these cells are adjacent and if the new cell on which the token lands has a strictly larger number than the old cell.
What is the minimum number of moves that Théodore can guarantee, whatever the way in which Maena has placed her integers?
Initially, Maéna numbers the cells of the grid from 1 to $99^{2}$, in an arbitrary way. Théodore then places a token on one of the cells of the square, and then allows himself moves of the following form: he may move the token from one cell to another only if these cells are adjacent and if the new cell on which the token lands has a strictly larger number than the old cell.
What is the minimum number of moves that Théodore can guarantee, whatever the way in which Maena has placed her integers?