# Advent calendar 2020

## 11 December

Noel has a large pile of cards. Half of them are red, the other half are black. Noel splits the cards into two piles: pile A and pile B.

Two thirds of the cards in pile A are red. Noel then moves 108 red cards from pile A to pile B. After this move, two thirds of the cards in pile B are red.

How many cards did Noel start with?

Note: There was a mistake in the original version of today's puzzle. The number 21 has been replaced with 108.

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Let's say there were originally \(t\) cards of each colour, and \(2a\) red cards and \(a\) black cards in pile A.
After the move, there were \(t-2a+108\) red cards and \(t-a\) black cards in pile B. Two thirds of the card in pile B are red, so:

$$t-2a+108=2(t-a)$$
$$t=108$$

Therefore there were 108×2=**216** cards.

Interestingly, it appears that this answer is independent of \(a\): any number of cards could be put in each pile and this situation would still work. However, only one situation could have happened
unless you allow there to at some points have been a negative number of cards in each pile.