Chapter 10: Brainteasers 383
at the same speed but in possibly different directions,
either clockwise or anticlockwise. When two ants meet
they immediately change directions, and then continue
with the same speed as before. Will the ants ever,
simultaneously, be in the same positions as when they
started out?
(Thanks to OMD.)
Solution
What are the chances of that happening? Surely all that
bouncing around is going to shuffle them all up. Well,
the answer, which you’ve probably now guessed, is
that, yes, they do all end up at the starting point. And
the time at which this happens (although there may be
earlier times as well) is just the time it would take for
one ant to go around the entire circle unhindered. The
trick is to start by ignoring the collisions, just think of
the ants walking through each other. Clearly there will
then be a time at which the ants are in the starting pos-
itions. But are the ants in theirownstarting positions?
This is slightly harder to see, but you can easily con-
vince yourself, and furthermore at that time they will
also be moving in the same direction they were to start
with (this is not necessarily true of earlier times at
which they may all be in the starting positions).
Four switches and a lightbulb
Outside a room there are four switches, and in the
room there is a lightbulb. One of the switches controls
the light. Your task is to find out which one. You cannot
see the bulb or whether it is on or off from outside the
room. You may turn any number of switches on or off,
any number of times you want. But you may only enter
the room once.
(Thanks to Tomfr.)