Puzzle : Two piles of equal cards facing up


I came across this extremely simple puzzle which looked a bit difficult initially.

Problem : You are in a dark room ( or completely blind folded ) and you are given a standard 52 card deck out of which 13 cards are facing up and rest all are facing downwards.  You have to divide the deck into two piles so that number of cards facing up in both the piles are equal. You are allowed to flip any card as many times as you want, however, you can’t tell whether a card is facing up or facing down by touching it.

Solution : Take any 13 cards and flip them all. Rest 39 cards will make another pile. These two piles will have the same number of cards facing up. Why? Well, say out of those 13 selected 0 \leq n \leq 13 cards are facing up.  Which means that in the pile with 39 cards there will be 13-n cards facing up. When you flip all 13 cards, number of cards facing up now becomes 13-n which is equal to what the other pile have.


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