J. O'Seph
by Werner Miller

Werner Miller is a retired teacher of mathematics, whose hobbies are recreational mathematics and magic. A magical inventor and writer, Werner has produced a prolific volume of work, mostly what he calls “semi-automatic” card tricks based on mathematical principles. He is the author of "Fast von selbst", "Alles Miller oder was", "Ratatouille", more than 300 trick contributions to various magazines and web sites and more than 30 related computer programs. In addition, Werner is also a staff member of the German magazine "Magische Welt", a columnist for the British magazine "The Magician", and regular contributor here at Visions. Werner's first English-language book, Ear-Marked, is available here.

Let's deal again with the "Josephus Problem".

Instead of continuously spelling "Y-E-S" or "N-O" as last month you may prefer to use other words, eg "R-E-D" or "B-L-A-C-K".
I can offer two solutions:

1. Have two 3-card packets dealt by the spectator. Ask the spectator to peek at the top card of either packet. After dropping the other packet on top, you mix the cards "Down/Under" style dealing the "down" cards side by side into a neat row. (The spectator's card will be the last one, ending up at the far right.)

Ask for the color of the noted card, but stress that the spectator is free to lie.

Whatever color is named, use it to locate openly the spectator's card by spelling. Start at the right end of the row and proceed to the left, thinking the row as an endless loop, and spell either "R-E-D, R-E-D, R-E-D,..." (eliminating every 3rd card) or "B-L-A-C-K, B-L-A-C-K, B-L-A-C-K,..." (eliminating every 5th card).
The last card will be automatically the card the spectator peeked at.

2. The spectator deals 11 cards back and forth into two equal packets. He notes and remembers the last card, places it on either packet and drops the other packet on top.

Give the combined packet an "Under/Down Deal", dealing the "down" cards into a row as in version 1. The spectator's card (originally at position no. 6) will end up third from the left.

Continue as in the 6-card version, but instead of eliminating the card falling on the last letter of the color discard the next one, ie every 4th card when you are spelling "R-E-D" and every 6th card when you are spelling "B-L-A-C-K".

Try it!

Postscript: If you want to know more about the "Josephus Problem", see eg W. W. Rouse Ball, Mathematical Recreations and Essays, London 1892, pp. 32-36, Maurice Kraitchik, Mathematical Recreations, New York 1942, pp. 93-94 or Geoffrey Mott-Smith, Mathematical Puzzles for Beginners and Enthusiasts, New York 1954, pp. 94-97.

Werner Miller