TITLE: A strange puzzle having two possible solutions
QUESTION [13 upvotes]: A friend of mine asked me the following question: 

Suppose you have a basket in which there is a coin. The coin is marked
  with the number one. At noon less one minute, someone takes the coin
  number one and put into the basket two coins: the number two and the
  number three. At noon less $\frac{1}{2}$ minute, he takes the coin
  number two and put into the basket the coins number $4$ and the number
  $5$ and so on. At noon less $\frac{1}{k}$ he takes the coin number $k$
  and put two other coins. 
The question is: How many coins I can find in the basket at noon?

Intuitively the answer is $\infty$ because I added one coin for infinite times. But the correct answer seems to be $0$ because every coin numbered $k$ has been removed at noon less $\frac{1}{k}$ of minute. What is the correct answer? Thanks.

REPLY [2 votes]: One possible way of intuiting a solution to this problem is found through "sizes" of infinity.  Intuitively, we feel that there are fewer odd natural numbers than there are natural numbers in total, because we can never fill the set of natural numbers purely from the set of odd natural numbers.
In the same way, the number of coins added to the basket is constantly larger than the number of coins removed.  No matter how often or how quickly you perform the task (and you head to infinitely fast as 1/k -> 0) you never remove as quickly as you add, meaning the coins added are a "larger" infinity than the coins removed.
The answer to what are the numbers on the coins are in the basket at noon, well "infinity" to "two infinity minus 1".  Which is clearly a nonsensical conclusion on the face of it, since no coin has such a label, but we can restate this to answer in a slightly better way:
At each time 1/k, we remove all coins in the basket with numbers less than k, and add new coins numbered 1 to k. We have the same outcome at each step k, except that instead of coins numbered k to 2k-1, they're now numbered 1 to k.  At noon, we will have a basket with an infinite number of coins numbered 1 to... well pick up the last coin and you'll know.