The Mathematics Of The Weave Shuffle

This article was originally presented in three consecutive issues of Pentagram, in mld-1957. (Errata for the article were published in the May 1958 issue of that journal.) It holds the distinction of being the first serious examination of the mathematics behind the faro shuffle to be published in magical literature. In his article, Mr. Elmsley took the seminal information presented in Hugard and Braue's Expert Card Technique (pp. 145-150, much of which was based on faro tables constructed by Fred Black), reinterpreted it mathematically, then expanded on it and contributed important new ideas to the subject. It was here, also, that he coined the terms "in-shuffle" and "out-shuffle", which have since become standard expressions in the trade.

Mathematical studies on the faro shuffle are more common today, though most are admittedly of far greater interest to the mathematician than to the magician. Over the years these studies have proven useful in statistics and set theory, and Mr. Elmsley's work was important. It is reproduced here as Mr. Elmsley originally presented it, with only minor editing. To the Pentagram monograph I have appended two related articles, one of which appeared a month after the Pentagram series, the other a year later, in P. Howard Lyons' Ibidem magazine.

While Mr. Elmsley's opening comments depreciate the usefulness of this information to magicians, it is suggested that the mathematically disinclined reader at least skim through it, particularly the sections titled "Out- and In-weaves", "The Odd Pack and Weave", "Equivalent Odd Pack", "Stack Transformations", "Royal Flush Deal", "The Restacking Pack", "Binary Translocations" and "Double Control". Practical magical applications are given in these sections, and one can see fascinating possibilities for further development. Indeed, much of this information has borne fruit for magicians since it was

Jlrst published. After this study we will turn to less theory-ridden discussions and some exceptional magic based on the faro shuffle and the principles detailed in this treatise.

This article is for mathematicians only. The many excellent tricks using the weave shuffle depend, with few exceptions, on the simplest of its properties. I ain going to deal with some of the more complicated and lesser known properties, most of which are useless to the magician, who is practical only, and I am writing primarily for those with an interest in mathematics quite apart from their interest in magic.

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