Revised to “Five or Six” from “Eight” November 7, 2010.<\/p>\n
In response to frequent questions, I now recommend shuffling the bridge deck just five or better six times and then preferably dealing the cards back and forth instead of cyclically. The recent article by Conger and Howald** supersedes the revolutionary 1992 paper of Bayer and Diaconis* in showing how the randomness of a shuffled deck is enhanced by dealing out the cards, even more so if the cards are dealt back and forth (West North East South South East North West) instead of the usual repeated cycle (West North East South West North East South). Their following table shows the remaining order after n shuffles for the undealt deck, for the bridge hands dealt cyclically as usual, and for the bridge hands dealt back and forth<\/p>\n
n \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 5 \u00a0 \u00a0 \u00a06 \u00a0 \u00a0 \u00a07\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 8 \u00a0 \u00a0 \u00a0 9 \u00a0 \u00a0 \u00a010<\/p>\n
undealt \u00a0 \u00a0 \u00a0 92% \u00a061% \u00a033% \u00a017%\u00a0\u00a0 8.5% \u00a04.3%<\/p>\n
cyclic deal \u00a0 23% \u00a0 7% \u00a0 3% \u00a0 \u00a0 2% \u00a0 \u00a0 1%<\/p>\n
back&forth \u00a031% \u00a0 3% \u00a0 1%<\/p>\n
Dealing back and forth has the added advantage of being a bit faster than dealing cyclically as usual. Some questions about the accuracy of the mathematical model remain.<\/p>\n
The Laws of Contract Bridge<\/a> and Duplicate Bridge<\/a> require “thorough” shuffling and that no two adjacent cards from the deck shall be dealt into the same hand (so that technically back and forth should use five piles).\u00a0The first recommends at least five shuffles. When the ACBL<\/a> started using computer shuffling, experts had to change their play. Previous inadequate shuffling had produced more ordered, flatter hands; tricks are collected in suits and dealt out evenly. Random hands are wilder. See Berger***.<\/p>\n *Dave Bayer and Persi Diaconis, “Trailing the dovetail shuffle to its lair,” Ann. Appl. Probab. 2 (1992), no. 2, 294-313.<\/p>\n **Mark A. Conger and Jason Howald, “A better way to deal the cards,” Amer. Math. Monthly 8 (Oct. 2010), 686-699. Won 2011 Mathematical Association of America Lester R. Ford Award.<\/p>\n