### Brain Teasers

# Cutting the Deck

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Fred opens a brand new deck of playing cards for a poker game and takes out the jokers so that just the 52 normal cards remain. Not knowing how to shuffle like the professionals, Fred decides to cut the deck at random depths for a number of times. (Cutting a deck is when 1 to 51 cards are taken off the top of the deck and placed under the remaining cards.) How many times must Fred cut the deck before he can be reasonably sure the deck is well shuffled?

A) 10 times

B) 52 times

C) 100 times

D) 2704 times (52^2)

E) Can't cut it enough.

A) 10 times

B) 52 times

C) 100 times

D) 2704 times (52^2)

E) Can't cut it enough.

### Hint

Try it on a smaller scale, say with 10 cards.### Answer

The answer is E) Can't cut it enough. No matter how many times you cut a deck of cards, the order of the cards remains the same with only the position of the cards changing. Cutting a deck of cards does not shuffle the cards at all. The top card may now be somewhere in the middle of the deck, but the card that was under it initially is still under it and the card that was under that card is still under it, etc.For example, let's just look at the 13 cards in one suit. The first card in the list is the top card and the last card is the bottom card. So before cutting, the cards look like this: (K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2). After a cut they may look like this: (7, 6, 5, 4, 3, 2, K, Q, J, 10, 9, 8). Cut it again and you may get something like this: (3, 2, K, Q, J, 10, 9, 8, 7, 6, 5, 4).

As you can see, the order remains the same no matter how many times they are cut, just their positions shift. In fact, 1 out of 52 times (about 1.9%) that Fred "shuffles" the deck in this way he will end up with the deck exactly as it was when he started.

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