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.)

Actually, doing the infinite series isn't that hard, as
you can see above.

I'm sure he was really smart, though.