Can you explain the answer ...?

Question

At one minute to midnight, a mathematician placed ten balls, numbered one through ten, into a vase and removed the ball marked with the number one.
* At one-half of a minute to midnight, the mathematician placed ten more balls, numbered eleven through twenty, into the vase and removed number two.
* At one-third of a minute to midnight, the mathematician placed ten more balls, numbered twenty-one through thirty, into the vase and removed number three.
* At one-fourth of a minute to midnight,...

At midnight, how many balls are in the vase?

My friend tell me that the answer is "zero". Can you explain ...

His argument is
"Name any ball and I will tell you it is not in the vase. Not only that, but I will tell you the exact moment at which it was removed from the vase "

My reasoning is:

"The mathematician places nine balls into the vase at each step; after an infinite number of steps, there must be an infinite number of balls in the vase."

but it does not give the same answer ...

Answer 1

Your reasoning is correct if in fact the placement of nine balls and the removal of one ball takes place in the infinite manner you have described.

There will always be 9 balls added for each ball removed. This goes on for infinity, so you might say there are infinite balls removed and (9 times infinity) balls remaining.

Your friend is either a genius beyond the likes of Einstein and Newton and Hawking: or he is dillusional thinking he does in fact know the answer; or he has misinterpreted the question and rationale; or quite simply he is wrong!

Answer 2

Well going by the information given, the mathematician keeps putting balls into the vase as midnight gets closer and closer. If he could technically keep performing this action faster and faster as the fraction of a second left got smaller and smaller, there would be an infinite number of balls in the vase since he would always be adding more into infinity.
The fact that he is removing a ball each time and the exercise lasts to the point of infinity, he would also remove an infinite number of balls.
So if he adds an infinite number and also removes an infinite number does that mean at midnight there are zero in the vase or infinite? He always adds more than he removes so saying "zero" hardly seems possible. If the answer is the latter that would mean you could have 2 kinds of infinite with one being greater than the other which hardly seems possible either. [as if this entire scenario is possible in the first place ;) ] The 2 amounts just approach infinity at different rates.

I guess you could just say there are 9 times as many balls in the vase as the number he removed to avoid having to go out and buy a lot of bingo balls and one gigantic vase.

Answer 3

Good question! I prefer your reasoning, because with every removal of one, you put in nine more, so with an infinite number of removals of one, you have nine times as many added, and they would outweigh the ones you take out. You would have an infinite number of balls in the jar.

I can see your friend's reasoning also, but it doesn't make sense to me, because no matter when you choose to examine the jar, there are always more balls in it than there were the instant before.

Questions involving infinity can be a real mental challenge, can't they?

Answer 4

is it because it was not actually 10 balls that mathematician put in the vase or is it that mathematician called the ball "1-10" and since the mathematician pulled out the ball that was the ball that was called "1" so the mathematician pulled tha ball call "1-10" because it had a one in it and it was the same with the rest of the balls that the mathematician put in there

Answer 5

You are both wrong. A quantum physicist would just calculate the number and tell it to you. When you arrive within the Planck time of midnight, the game stops, because time can no longer be subdivided! Sometimes you have to love quantum physics!

Answer 6

Whatever.