Question 29
A $2013\times2013$ board has north–south and east–west grid lines. Each cell contains an arrow pointing east or south. During each minute, for every arrow $A$, if $A$ points to an adjacent cell whose arrow $B$ has a different direction, change $A$ to the other direction for the next minute (south becomes east and east becomes south); otherwise leave $A$ unchanged. All these changes are simultaneous. Prove that after finitely many minutes no arrow changes direction, and determine the latest possible minute in which at least one arrow can change.