Two bicyclists start 100 miles apart, and head towards each other, each
one going 10 mph. At the same instant, a fly leaves the first bike and
flies at 20 mph to the second. When it gets there, it immediately turns
around and heads back to the first. Then it repeats, going back and forth
between the two bikers. By the time they reach each other, how far will
the fly have travelled?
The easy way to solve this problem is to realize that the bikers are
approaching each other at a net speed of 20 mph, so it will take 5 hours
for them to meet. During that time the fly is travelling constantly,
back and forth, at 20 mph, so in 5 hours it will travel exactly 100 miles,
and that is the answer.
The harder way is to sum an infinite series. The first leg has the fly
meeting the biker at a net speed of 30 mph. They start off 100 miles
apart so it takes 3 1/3 hours to meet, during which the fly travels 66 2/3
miles. Now the fly will turn around. By this time the bikers have
drawn to 33 1/3 miles apart. So this second leg of the trip will take
1/3 as long. Similarly the third leg will take 1/3 of the second, or 1/9
of the first, and so in indefinitely, with each leg taking 1/3 as long
(and hence the fly going 1/3 as far) as the previous one. So the answer
will be 66 2/3 times (1 + 1/3 + 1/9 + 1/27 + ...). The value of that
infinite series is exactly 1 1/2, so the answer is 66 2/3 times 1 1/2,
or 100 miles.
Von Neumann thought for a brief moment and gave the answer. The hostess
was disappointed and said, oh, you saw the trick, most people try to sum
the infinite series. Von Neumann looked surprised and said, but that's
how I did it. (Cue laughter/applause.)
