Question 47
On an $m \times m$ board, at the midpoints of the unit squares there are some ants. At the time $0$ each ant starts moving with speed $1$ parallel to some edge of the board until it meets an ant moving in the opposite direction or until it reaches the edge of the board. When two ants moving in the opposite direction meet each other, both turn $90^\circ$ clockwise and continue moving parallel to another edge of the board. Upon reaching the edge of the board the ant falls off the board.
a) Prove that eventually all the ants will have fallen off the board.
b) Find the latest possible moment for the last ant to fall off the board.
a) Prove that eventually all the ants will have fallen off the board.
b) Find the latest possible moment for the last ant to fall off the board.