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?