Stack Transformations

Suppose, hi an odd pack of P cards, we have a set of cards stacked "a" apart, counting downward through the pack. As we shuffle, the separation of the cards in the stack becomes successively 2a, 4a, 8a and so on. Suppose, after x shuffles, the separation of the cards in the stack (which is 2xa) is less than P but greater than half P. If we now count in the opposite direction; i.e., upward through the pack, the separation of the cards in the stack is P - 2*a. This may give an interesting transformation of one stack to another. At each transformation the order of the cards in the stack reverses.

With an even pack, the calculations are made as for the equivalent odd pack.

For example, fifty-two cards out-shuffled are equivalent to an odd pack of fifty-one. Therefore, when a full pack is given successive out-shuffles, a stack of cards seven apart within it will move farther apart in this pattern: 7, 14, 28 or 51 -28 = 23, 46 or 51 - 46 = 5.

Thus a stack seven apart becomes a stack five apart after three out-shufiles. There were two transformations, so the stack is in its original order. If, after shuffling, the distance between two cards of the stack has to be counted round the ends of the pack, there will be an error due to our using an even pack.

Other transformations with a pack of fifty-two out-shuffled are:

10 to 11 in two shuffles, reversing their order.

11 to 7 in two shuffles, reversing their order.

Any multiple of three to itself in four shuffles, reversing their order.

With fifty-two cards in-shuffled we get:

10 to 13 in two shuffles, reversing their order.

11 to 9 In two shuffles, reversing their order.

7 to 3 in three shuffles, retaining their order.

3 to 5 in four shuffles, reversing their order.

The 10 to 13 transformation means that cards ten apart can be brought together by four in-shuffles.

Here, incidentally, is a poker deal that uses the 7 to 3 transformation.

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