## The Restacking Pack

We know that fifty-two cards return to their original order after eight out-shuffles. During these shuffles the top and bottom cards never move, and two more, the eighteenth and thirty-fifth cards, change places in each shuffle. The remaining forty-eight cards can be considered as lying in six chains, each chain comprising eight positions. A card at any of these positions will move through each of the other positions in its chain before returning to its original position after eight shuffles.

If we place four cards of the same value at alternate positions in a chain, after two shuffles each of the four cards will have moved into a position previously occupied by one of the others. For example, one chain is 2-3-5-9-17-33-14-27. If we place the four jacks at 2, 5, 17 and 14, after two shuffles there will still be jacks at each of these positions.

Hence, if we place four cards of the same value at alternate positions, in all six chains, we shall have arranged twelve sets of four cards in forty-eight positions in such a way that, after two shuffles, every one of the forty-eight positions will be occupied by a card identical in value to the card that was originally at that position. If the remaining four cards of a kind go on top, eighteenth, thirty-fifth, and on the bottom, each of these cards will be in the same position after two shuffles.

Thus we now have a stacked pack that, disregar ding suits, returns to the same order after two out-shuffles, (As an aside, It is worth noting that, with such a stack, the ordering of values created on the odd out-shuffles, though different from the original order, also remains constant with each alternate shuffle. That is, the sequence of values found after one out-shuffle will be repeated after the third such shuffle, and the fifth, etc.)

Here is an example of such a stack. The cards are arranged to be read from left to right, moving from the top of the deck to the bottom:

 2c jc 0h 6s jd 10d 4s qh 9c 5s qd 8c 6h jh 10c 5d js 2h 3c kh 10s 3d ad 8s 4h kc 9s 4d qc 10h 3h ac 9d 3s 2s 7c 5h as 7d 6d qs 8h 5c iîs 8d 4c ah 7h 6c kd 7s 2d ¥ * * * * * * ¥ * * * * * * ¥ ¥

The suits have been so arranged that, apart from the twos, after two shuffles the position previously occupied by any card will be occupied by a card of the same value, but one suit advanced in the rotation clubs-hearts-spades-diamonds. The twos will always be in the same positions.

It will be seen that the cards fall into three groups of sixteen, separated by the twos. The suit order In each group of sixteen is the same. If each group of sixteen is divided into four sets of four, each set contains four cards of different suits. The suit orders in the second and thu d sets are respectively the reverse of the orders in the first and fourth sets.

The card values fall into four classes, each comprising three values. The division of the values among the classes is, of course, arbitrary, but in the stack given above the classes are 3-7-J, 4-8-Q, 5-9-K and 6-10-A. The values in any one class are always found at the same positions in the groups of sixteen.

It hardly needs saying that you can obtain other restacking stacks from this stack by interchanging cards of one value for cards of another. (Edward Mario published some ideas on Mr. Elmsley's restacking pack. These can be found In Mr. Mario's Faro Controlled Miracles, pp. 18-19: and again in Alton Sharpe's Expert Card Mysteries, pp. 175-178. S.M.|

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