"The problem starts with an empty vase and an infinite supply of balls at a starting time before noon. At each step in the procedure, balls are added and removed from the vase. The question is then posed: How many balls are in the vase at noon?
At each step, balls are inserted into and removed from the vase in a particular order:
* In the first step, ten balls (numbered 1 through 10) are added to the vase, and then the first ball (numbered 1) is removed from the vase.
* At each subsequent step, ten more balls are added to the vase (numbered 10(n−1)+1 through 10(n−1)+10 at step n), and then the lowest numbered ball (n) is removed from the vase.
As part of the problem statement, it is assumed that an infinite number of steps is performed. This is allowed by the following conditions:
* The first step is performed at one minute before noon.
* The second step is performed at 30 seconds before noon.
* The third step is performed at 15 seconds before noon.
* Each subsequent step n is performed at 2−(n−1) minutes before noon.
This guarantees that a countably infinite number of steps is performed by noon."
At each step, balls are inserted into and removed from the vase in a particular order:
* In the first step, ten balls (numbered 1 through 10) are added to the vase, and then the first ball (numbered 1) is removed from the vase.
* At each subsequent step, ten more balls are added to the vase (numbered 10(n−1)+1 through 10(n−1)+10 at step n), and then the lowest numbered ball (n) is removed from the vase.
As part of the problem statement, it is assumed that an infinite number of steps is performed. This is allowed by the following conditions:
* The first step is performed at one minute before noon.
* The second step is performed at 30 seconds before noon.
* The third step is performed at 15 seconds before noon.
* Each subsequent step n is performed at 2−(n−1) minutes before noon.
This guarantees that a countably infinite number of steps is performed by noon."
The answer is zero. For every ball n, it is subsequently removed in step n; thus in the end, there is no ball that can be said to remain in the vase.
If you graph it, there'd be a jump discontinuity where time is 12:00.
My mind is blown.
Part 2:
crobert says: hilbert's grand hotel ftw
Peter says: I heard of that
Peter says: sucks for the first person
crobert says: yeah
Peter says: so uh
Peter says: instead of the previous tenents moving down
Peter says: why don't the newcomers just go to the end?
crobert says: you could do that
crobert says: it'll end up being the same
Peter says: it seems more fair though
Peter says: because the late arrivals are the ones
Peter says: who have to find room lim -> infinity
crobert says: no person i is moved to room (i + 1)
Peter says: but assuming the hotel is full
Peter says: then isn't i lim -> infinity?
crobert says: but by definition you can always find one more room
crobert says: because there are infinite rooms
Peter says: but whole idea
Peter says: was that they had a hotel with infinite rooms
Peter says: that's full
Peter says: how do they accomodate more guests
Peter says: oh shi-
crobert says: gg
Peter says: that's why everyone had to move down
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