Puzzle 9: Problems with ants!

The noted gourmet Pangolini Aardvark is preparing a late night snack of "Ant au Chocolat" and "Ant au Fromage". This requires the use of a five foot pole. One end of the pole is over a bucket of melted chocolate and the other is over a bucket of melted cheese.

Pangolini sprinkles some ants onto the pole. They immediately start scampering along the pole in random directions. If two ants run into each other then they both instantaneously reverse their directions and are now moving away from each other. An ant can change direction many times. Eventually, all of the ants will fall off one or other end of the pole. If each ant travels at a speed of one inch per second, what is the maximum time until all ants have fallen off?

Suppose now that n ants are placed on a circle of five foot circumference and randomly choose their direction of travel and again reverse direction when they bump into each other. One of the ants is named Alice. What is the probability that Alice is back where she started, one minute after the ants start their scampering.

Back to the pole. Alice starts in the middle of the pole. There are n other ants placed randomly on the pole and they start scampering in random directions. Alice has a cold. When an ant with a cold bumps into another ant, the uninfected ant catches a cold too. What is the expected number of ants who catch cold before they all fall off the pole?


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