While investigating the effect of combined in- and out-shuffles, I fell into the practice of abbreviating them as T and 'O*, This led me to the discovery of a fortunate coincidence, for I noticed that my sequences of Ts and 'O's could be read as Ts and 'O's; and these could be manipulated with binary arithmetic to yield useful instructions for shuffle sequences. Here are several applications I've derived from this principle.
1) To bring the top card of the pack to any position, subtract one from the desired position, express the result as a binary number and use it as instructions for a series of in- and out-shuffles.
Example: If you wish to move the top card to a position fifteenth from the top, first subtract 1 from 15, getting 14. Write 14 in binary notation: 1110. Interpreting the 'l's for 'I's and the 'O's for 'O's, perform three in-shuffles followed by one out-shuffle. The card originally on top is now fifteenth from the top. This is the shortest possible number of shuffles that will perform the desired translocation. This method of generating shuffle sequences holds for decks of any size, odd or even.
2) In a pack of 2X cards, to bring a card at a known position to the top. subtract one from its position, express the remainder as a binary number, add zeros in front of it if necessary to make it a number of x figures, and use the result as a pattern for shuffling. If the result ends in one or more zeros, these can be ignored, since an out-shuffle retains the top card. [In addition, as Ravelli pointed out in Ibidem, No. 14 (p. 7), the final ln-shuffle need only be correct for the first cards of the packets. Indeed, a simple cut at center, rather than a shuffle, can be made and completed at this point. S.M.]
Example: In a pack of 32 cards (32 = 25; therefore x = 5), to bring the fifteenth card to the top:
x = 5, so we add one *0' to the front of the binary number to bring it to five figures: 01110. Since we can ignore the final zero we get 0111. Therefore, one out-shuffle, followed by three in-shufiles brings the fifteenth card to the top.
3) If you have an edge-marked card in a pack of 2X cards, it can be brought to the top in x or fewer shuffles, by always in-shuffling when the marked card is in the bottom half and out-shuffling when it is in the top half.
I have so far been unable to discover a comparatively simple way of bringing a card to the top of a deck that is not a power of 2; e.g., 52. The only method I have found is much too complicated for practical use. However, if you perform reverse faro shuffles instead of faros, a card at any position In any size pack can be brought to the top by subtracting one from its position in the pack and translating the remainder into a binary number. The result provides a sequence for reverse faro shuffling that will bring the card economically to the top, when read in reverse; that is, from right to left.
[Mr. Elmsley's statement of this problem encouraged others to attempt solutions. David Michael Evans provides this list of references, which offers information and solutions of varying merit: "Something from Ravelli of Switzerland", Ibidem, No, 13 (March 1958), p. 10; Marlo'sFaro Notes (1958), pp. 2-6; "Oil Always Floats" in Swinford's More Faro Fantasy (1971), pp. 54-57; Murray Bonfeld's "A Solution to Elmsley's Problem" in Genii, Vol. 37, No. 5 (May 1973), pp. 195-196; and "Any Card, Any Number" in Bonfeld's Faro Concepts (1977), pp. 41-47. S.M.)
June, July, August, September 1957
Here is an interesting, though uninspired, magical application of the rule last explained. It uses a thirty-two card deck. You might reduce a fifty-two card deck to thirty-two by first performing a trick that ends with a four-hand poker deal. Push the four hands aside at the finish and continue with the balance of the pack.
Have a card chosen and noted. When it is returned, use your nail to scratch or nick it secretly on one long edge. Hand the deck to the spectator for shuffling. When he has finished, have him note the position at which his card now lies from the top.
Take the pack from him and ask a second spectator to remember the top card. Then hold the pack with the marked edge of the card nearest you and perform five faro shuffles. In-weave whenever the edge-marked card is in the lower half of the pack; and out-weave whenever it is in the upper half. When you have finished shuffling, ask for the name of the chosen card. Turn over the top card of the pack: it is the selection. Turn the card facedown again and ask for the number at which the card rested before you shuffled. Count down to that number and turn up the card there. It will be the card remembered by the second spectator, which was on top before the shuffles.
[For a var iant on this idea, see "Elmsley Revisited" in Swinford's More Faro Fantasy, p, 63. S.M.]
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