Josephus Words

Article excerpt

In the articles on "Magic Spells" (Word Ways, Feb and May 2010) it was proposed that tricks could be performed with a deck of n letter cards. The deck would be prearranged, spelling some word u. There would be a "skip sequence" of integers [k.sub.1], [k.sub.2], ..., [k.sub.n]; the more natural the sequence the better. The magician would spell a new word w = [w.sub.1][w.sub.2][w.sub.3] ... [w.sub.n] as follows: skip [k.sub.1] cards and set the next card aside making it [w.sub.1], skip [k.sub.2] cards and set the next card aside making it [w.sub.2], etc. Each skipped card is returned to the bottom of the deck. Note that the skip sequence defines a permutation [pi] of the the original deck order; w = [pi] (u). We say w is a fixed-point if w = [pi](w). For any given permutation there exists a skip sequence, though it might be hard for a magician to incorporate.

The logological question is to find pairs of words u and w, and a well-motivated skip sequence relating them, that a magician could use with suitable patter. I am not a magician, however, so in this article I will just give pairs of common words. (Pairs using an uncommon word were found but are not reported.)

The story of Josephus Flavius is well-known in recreational mathematics. Forty men stood in a circle and every third man, still standing, was killed. (The puzzle is to find where Josephus should stand to survive to the end.) In our terminology we would say [k.sub.i] = 2 for all i. However [k.sub.1] might be different depending on where you want to start. Let [J.sup.b.sub.a] be the skip sequence where [k.sub.1] = a and [k.sub.i] = b for i > 1. Choosing a = 0 or a = b would be natural in a trick. …