This puzzle was originally told to me by a guy in Valhalla, the grad
student bar at Rice University. Since then, I've told it whenever
I thought it was even marginally on-topic. This has resulted
in me being accused of proof-by-intimidation and generally being
disliked.
## Question 1:

1. Remove the 10 lowest numbered balls from the first
bucket.

2. Throw away the lowest numbered ball from that group.

3. Place the remaining nine in the second bucket.

4. Repeat steps 1-3 until the first bucket is empty.

Example: you remove ping pong balls numbered 1 through 10, throw away ping pong ball number 1, and put numbers 2 through 10 in the second bucket. Then remove 11 through 20, throw away 11, and put 12 through 20 in the second bucket.

**When the first bucket is empty, how many balls in the second bucket?**
## Question 2:

(Remember we are starting with a new full and empty bucket).

1. Remove the 10 lowest numbered balls from the first bucket.

2. Put them in the second bucket.

3. Remove and throw away the lowest numbered ball from the second bucket.

4. Repeat steps 1-3 until the first bucket is empty.

Example: you remove ping pong balls number 1 through 10 from the first bucket, put them in the second bucket, and remove and throw away ping pong ball number 1. Remove 11 through 20, put them in the second bucket, and throw away ping pong ball 2.

**When the first bucket is empty, how many balls in the second bucket?**
### Question 3 (bonus):

This is the same as question 2, except instead
of throwing away the lowest numbered ball, we'll throw away a ball
chosen at random.

**When the first bucket is empty, how many balls in the second bucket?**

All three questions start with the same scenario: you have two very large (infinitely large) buckets. In one bucket, you have a very large (infinitely large) number of ping pong balls, each one has a number painted on it. They start at 1, 2, 3 and go up forever. No ping-pong ball has an omega painted on it. Each question involves pulling balls out of the filled bucket and putting them in the empty bucket.

2. Throw away the lowest numbered ball from that group.

3. Place the remaining nine in the second bucket.

4. Repeat steps 1-3 until the first bucket is empty.

Example: you remove ping pong balls numbered 1 through 10, throw away ping pong ball number 1, and put numbers 2 through 10 in the second bucket. Then remove 11 through 20, throw away 11, and put 12 through 20 in the second bucket.

1. Remove the 10 lowest numbered balls from the first bucket.

2. Put them in the second bucket.

3. Remove and throw away the lowest numbered ball from the second bucket.

4. Repeat steps 1-3 until the first bucket is empty.

Example: you remove ping pong balls number 1 through 10 from the first bucket, put them in the second bucket, and remove and throw away ping pong ball number 1. Remove 11 through 20, put them in the second bucket, and throw away ping pong ball 2.

If you think you know the answers, consider this: each question has a different answer. For example, if the answer to the first question is 4 (it isn't), neither the second nor third question's answer is 4.

If you enjoyed this, check out Hilbert's Hotel.