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.
Well, for each step, one net ball is added and one net ball is dropped. Since he has infinite balls to start with, then the number in his bucket is infinity / 2, as is the number that dropped out. Infinity being what it is, there are an infinite number of balls both in and out of the bucket!
Originally posted by sven1000'nuff said
Well, for each step, one net ball is added and one net ball is dropped. Since he has infinite balls to start with, then the number in his bucket is infinity / 2, as is the number that dropped out. Infinity being what it is, there are an infinite number of balls both in and out of the bucket!
Originally posted by sven1000This is the answer most people come up with -- but it is not correct.
Well, for each step, one net ball is added and one net ball is dropped. Since he has infinite balls to start with, then the number in his bucket is infinity / 2, as is the number that dropped out. Infinity being what it is, there are an infinite number of balls both in and out of the bucket!
Originally posted by davegageEach ball went in the bucket once and fell out once, so the bucket contains no balls after two units of time.
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 ...[text shortened]... how many ping pong balls remain in the bucket?
*No one but Chuck could accomplish this feat.
Alternatively, consider the set of balls in the bucket when he finishes. Since it's a subset of a countable set of balls, it has an element with least number if it's nonempty. Call this ball B (B is some positive integer) and follow the algorithm. It's pretty clear that Ball B can't exist, so the set has no minimal element and is thus empty.
Originally posted by davegageAll the balls.
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 ...[text shortened]... how many ping pong balls remain in the bucket?
*No one but Chuck could accomplish this feat.
He stuffs the hole shut with Jack Bauer and Macgyver in the first unit of time, and Roundhouses all the balls into the bucket in half a time unit.
He uses the last half a unit to save a mother and her child from thugs.
P-
Originally posted by PhlabibitWell, I was going to say that royalchicken is right on the money with his proof by contradiction that ball B is non-existent; I was also going to add that another way to see that royalchicken must be right is to note simply that the sets {ball 1, ball 2, ... , ball n, ... } and {(ball 1, ball 2), (ball 3, ball 4), ... , (ball 2n-1, ball 2n), ... } are equipolent (one-to-one correspondence).
All the balls.
He stuffs the hole shut with Jack Bauer and Macgyver in the first unit of time, and Roundhouses all the balls into the bucket in half a time unit.
He uses the last half a unit to save a mother and her child from thugs.
P-
But now that I have seen your answer and explanation, I think we all need to re-evaluate our core beliefs. Well done, Phlabibit!
Originally posted by PhlabibitThis is very insightful reasoning 😀!
All the balls.
He stuffs the hole shut with Jack Bauer and Macgyver in the first unit of time, and Roundhouses all the balls into the bucket in half a time unit.
He uses the last half a unit to save a mother and her child from thugs.
P-