03 Jun '06 22:58

Suppose Chuck Norris* has infinitely many ping pong balls (the balls are labeled ball 1, ball 2, ball 3, ... , ball n, ... ). He also has a magic bucket with a special hole in the bottom.

The clock starts at time t = 0. At time t = 1, Chuck places ball 1 and ball 2 into the bucket, and ball 1 instantaneously falls out the special hole, leaving ball 2 in the bucket. At time t = 1 + 1/2, Chuck places ball 3 and ball 4 into the bucket, and ball 2 instantaneously falls out the special hole, leaving ball 3 and ball 4 in the bucket. At time t = 1 + 1/2 + 1/4, Chuck places ball 5 and ball 6 into the bucket, and ball 3 instantaneously falls out the special hole, leaving ball 4, ball 5, and ball 6 in the bucket. I think you get the idea here: generally, at time t = 1 + 1/2 + 1/4 + ... + 1/2^m, Chuck places two more balls into the bucket, and the lowest numbered ball in the bucket falls out of the bucket through the hole. Since SUM(m=0 to m=M)[1/2^m] converges to 2 as M --> infinity, after 2 units of time, Chuck is finished placing all of his ping pong balls into the bucket.

So: after 2 units of time, how many ping pong balls remain in the bucket?

*No one but Chuck could accomplish this feat.

The clock starts at time t = 0. At time t = 1, Chuck places ball 1 and ball 2 into the bucket, and ball 1 instantaneously falls out the special hole, leaving ball 2 in the bucket. At time t = 1 + 1/2, Chuck places ball 3 and ball 4 into the bucket, and ball 2 instantaneously falls out the special hole, leaving ball 3 and ball 4 in the bucket. At time t = 1 + 1/2 + 1/4, Chuck places ball 5 and ball 6 into the bucket, and ball 3 instantaneously falls out the special hole, leaving ball 4, ball 5, and ball 6 in the bucket. I think you get the idea here: generally, at time t = 1 + 1/2 + 1/4 + ... + 1/2^m, Chuck places two more balls into the bucket, and the lowest numbered ball in the bucket falls out of the bucket through the hole. Since SUM(m=0 to m=M)[1/2^m] converges to 2 as M --> infinity, after 2 units of time, Chuck is finished placing all of his ping pong balls into the bucket.

So: after 2 units of time, how many ping pong balls remain in the bucket?

*No one but Chuck could accomplish this feat.