For instance, the set of all configurations of a deck of 52 cards forms a group, and a shuffle is an operation on that group. Group theory concerns itself with understanding sets and properties preserved by operations on those sets. This fact may come as somewhat of a surprise, because there are 52! possible deck configurations, and since there is no randomness, after 52! out-shuffles, we must hit some configuration at least twice (and then cycle from there). card trick and see if they can figure out the rest: whether 0 or 1 stands for an in/out shuffle, and whether to read the digits from left to right or vice versa. Notice the 2nd, 3rd, and 4th hang over the edge, this is important. card trick and see if they can figure out the rest: whether 0 or 1 stands for an in/out shuffle, and whether to read the digits from left to right or vice versa. Riffle Shuffle In The Hands With your right hand grip the deck from above, thumb at the short inner edge, 1st finger curled on top, 2nd and 3rd at the short outer edge, 4th (pinky) at the right longer edge. As a project, you might even tell them part of the binary card trick and see if they can figure out the rest: whether 0 or 1 stands for an in/out shuffle, and whether to read the digits from left to right or vice versa. (Answer: 52.) You can also have students investigate decks of smaller sizes. Have students go home and determine how many in-shuffles it takes to restore the deck to its original order.
Since 6 is 110 in binary notation, then the sequence IN-IN-OUT will move the top card (position 0) to position 6 (the seventh card). Voila! You will now have the top card at position N. Perform an out-shuffle for a 0 and and in-shuffle for a 1. Write N in base 2, and read the 0's and 1's from left to right. Surprise: 8 perfect out-shuffles will restore the deck to its original order!Īnd, in fact, there is a nice magic trick that uses out and in shuffles to move the top card to any position you desire! Say you want the top card (position 0) to go to position N. The in-shuffle is one in which the top card moves to the second position of the deck. There are 2 kinds of perfect shuffles: The out-shuffle is one in which the top card stays on top.
What happens if you do perfect shuffles over and over again? We know from the Fun Fact Seven Shuffles that 7 random riffle shuffles are enough to make almost every configuration equally likely in a deck of 52 cards.īut what happens if you always use perfect shuffles, in which you cut the cards exactly in half and perfectly interlace the cards? Of course, this kind of shuffle has no randomness.
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