Ants on a Stick

The problem
1 m stick · ants at 1 m/s · reverse on collision0 m1 mlongest time until all have fallen off?

A bunch of ants sit on a one-meter stick at random positions, each facing left or right. They all walk at exactly 1 m/s. Rules:

  • When two ants meet head-on, both reverse direction.
  • When an ant reaches an end, it falls off.

In the worst case (any number of ants, any positions, any starting directions), how long until all ants have fallen off?

Tempting (but wrong)
cascading reversals · feels unboundedpeople reach for simulation, ODEs, case analysis

The collisions look like they could compound indefinitely. With many ants, you get cascading reversals — an ant turns around, hits another, turns around again. Tracing this seems to require simulation or careful case work, and the answer feels like it should grow with the number of ants.

It doesn't.