Question 1
A table consists of an $m\times m$ array of unit squares. Some square centres initially contain one ant each. From time $0$, every ant moves at speed $1$ parallel to a side of a square. When ants meet:
(i) If exactly two meet head-on, both turn clockwise through $90^\circ$ and continue at speed $1$.
(ii) If two meet at right angles, or more than two meet, they keep their velocities and directions.
An ant reaching the edge falls off and never returns. The time when the last ant falls is called their “doomsday”. Over all possible initial placements, find the latest possible doomsday, or prove that doomsday need not occur.
(i) If exactly two meet head-on, both turn clockwise through $90^\circ$ and continue at speed $1$.
(ii) If two meet at right angles, or more than two meet, they keep their velocities and directions.
An ant reaching the edge falls off and never returns. The time when the last ant falls is called their “doomsday”. Over all possible initial placements, find the latest possible doomsday, or prove that doomsday need not occur.