Question 14
All cells of an $n\times n$ square table are numbered in some order from $1$ to $n^2$. Petya makes moves according to the following rules. On his first move he places a token in any cell. On each subsequent move he may either place a new token in a cell or move a token horizontally or vertically from a cell numbered $a$ to a cell with a larger number. Whenever a token enters a cell, that cell is immediately colored; placing a token on a colored cell is forbidden. What is the smallest number of tokens Petya needs to be able, regardless of the initial numbering, to color all cells in finitely many moves?
(D. Khramtsov)
(D. Khramtsov)