Wealthy banker has three daughters and infinitely many 1 dollar bills, serial numbers 1,2,3, ... n...?

Question

Each daughter demanded infinitely many dollar bills for christmas shopping spree. The banker knows, that neither investors nor his daughtres check serial numbres of the bills, they are concerned only about total amount.

How can the banker embezzle enough money to please all his daughters, and get away with it during the next audit?

Answer 1

Why worry, you can give away an infinite amount of money and still have an infinite amount. How are those accountants going to know the difference? Infinity divided by 4 is still infinite (1 share for each daughter and one share for the bank).

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But then again, if you are going to have serial numbers on all the bills, are they truly infinite?

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Edit: I'm going to make an assumption here: The investors do the audit. If an accountant did the audit then he/she would probably want a random sample of bills to ensure that none was missing. Particularly when dealing with an infinite amount. :-)

Since the investors are doing the audit and they don't care about the serial numbers, we are in luck.

The serial numbers compose the set of natural numbers. This is a countable infinite. We just got to create a mapping function to divide everything out.

Daughter 1 gets all the prime numbers, Daughter 2 gets all the prime numbers *2 (excepting 2, of course) and Daughter 3 gets all the prime numbers *3 (excepting 2, 3 and 6). The Investors get the rest. Infinity divided 4 ways and all sets are infinite!

Answer 2

Imagine that the bank has all the serial numbers of its infinitely many dollar bills arrayed in a finite 3D spherical ball. Then, by the Banach-Tarski Theorem, this sphere can be partitioned into 5 distinct pieces such that, when reassembled, produces 2 spheres identical to the first. Let the bank keep one, and give the other to the daughters.

Answer 3

mary them to rich husbands