MathDB

Let

m,n\in \mathbb{Z}_{\ge 1}

and a rectangular board

m\times n

sliced by parallel lines to the rectangle's sides into

mn

unit squares. At moment

t=0

, there is an ant inside every square, positioned exactly in its centre, such that it is oriented towards one of the rectangle's sides. Every second, all the ants move exactly a unit following their current orientation; however, if two ants meet at the centre of a unit square, both of them turn

180^o

around (the turn happens instantly, without any loss of time) and the next second they continue their motion following their new orientation. If two ants meet at the midpoint of a side of a unit square, they just continue moving, without changing their orientation.\\ \\ Prove that, after finitely many seconds, some ant must fall off the table.\\ \\ (Oliver Hayman)

Problem Statement