## Five or Six (not Eight) Shuffles

Revised to “Five or Six” from “Eight” November 7, 2010.

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

n                 5      6      7       8       9      10

undealt       92%  61%  33%  17%   8.5%  4.3%

cyclic deal   23%   7%   3%     2%     1%

back&forth  31%   3%   1%

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.

The Laws of Contract Bridge and Duplicate Bridge require “thorough” shuffling. The first recommends at least five shuffles. When the ACBL 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***.

*Dave Bayer and Persi Diaconis, “Trailing the dovetail shuffle to its lair,” Ann. Appl. Probab. 2 (1992), no. 2, 294-313.

**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.

***Paul D. Berger, On the distribution of hand patterns in bridge: man-dealt versus computer-dealt, Canadian J. Stat. 1 (1973), 261-266.

1. #### Stu Casper:

Thanks for the info on shuffling.
I worry about computer shuffling when a computer deals out many thousands of hands a day.
It is my belief that the cpu capacity limits the number of random deals, and then the deals repeat.
In BridgeBaseONline there are thousands of hands dealt daily.
I would like to see you computer/math wizards explore this fully.

2. #### Tom:

Correction: In 1st (long) paragraph, you obviously meant “after n shuffles”, not “after n deals”.

Thanks, fixed.

I assume your model assumes an ideal shuffle (1 to 1 interlap). The reality is obviously quite different, although how this affects the conclusion isn’t.

Actually the model realistically posits a random interlap. Perfect shuffles do not mix cards well; after eight of them the deck is back in the original order. —FM

3. #### Thomas Quinlan:

What of the article titled How to Win at Poker under Science and Technology in The Economist of Oct 12, 1996. A seemingly erudite discusson of the mathematics of card shuffling. It seems to talk about 7 shuffles for a random mixing. It also discusses the outshuffles and inshuffles.

Yes, as my table shows, it takes at least 7 or 8 shuffles to randomize an undealt deck, but with the further randomization provided by dealing, 5 shuffles suffices. — FM