From: jsavard@fNrOeSePnAeMt.edmonton.ab.ca (John Savard) Subject: Re: 'Scientific American' issue... which one? Date: Tue, 05 Sep 2000 06:28:14 GMT Newsgroups: sci.math Summary: [missing] On Mon, 04 Sep 2000 21:14:53 -0700, Fred M. Sloniker wrote, in part: >The >bit I particularly remember involved three six-sided dice which were >numbered unusually; die A would tend to roll higher than die B, which >would tend to roll higher than die C, *which would tend to roll higher >than die A*. I don't remember the issue, but I can tell you how the dice worked. Let's start with two dice, with faces: 5 5 5 5 1 1 and the other with faces 6 6 2 2 2 2 The first die will beat the second die 4/9 of the time, and lose the rest of the time. So the second die is better. But a die that shows either 3 or 4, in any mix, will beat the second die 2/3 of the time, and be beaten by the first die 2/3 of the time. John Savard http://home.ecn.ab.ca/~jsavard/crypto.htm ============================================================================== From: jsavard@fNrOeSePnAeMt.edmonton.ab.ca (John Savard) Subject: Re: 'Scientific American' issue... which one? Date: Tue, 05 Sep 2000 06:34:08 GMT Newsgroups: sci.math On Mon, 04 Sep 2000 21:14:53 -0700, Fred M. Sloniker wrote, in part: >The >bit I particularly remember involved three six-sided dice which were >numbered unusually; die A would tend to roll higher than die B, which >would tend to roll higher than die C, *which would tend to roll higher >than die A*. Here are the URLs of some sites which talk about this kind of set of dice: http://www.geocities.com/CapeCanaveral/Hangar/7773/dice.html http://exploringdata.cqu.edu.au/nt_dice.htm http://www.maa.org/mathland/mathtrek_10_6.html John Savard http://home.ecn.ab.ca/~jsavard/crypto.htm ============================================================================== From: Gerry Myerson Subject: Re: 'Scientific American' issue... which one? Date: Wed, 06 Sep 2000 09:46:48 +1000 Newsgroups: sci.math In article , Fred M. Sloniker wrote: > In one of the recent 'Scientific American' issues, there was an > article that talked about probabilities and non-standard dice. It may be reprinted as one of these: Gardner, Martin. 1987. Nontransitive paradoxes. In Time Travel and Other Mathematical Bewilderments. New York: W.H. Freeman. ______. 1983. Nontransitive dice and other probability paradoxes. In Wheels, Life, and Other Mathematical Amusements. New York: W.H. Freeman. Consider the magic square 8 1 6 3 5 7 4 9 2 A die with faces 8, 1, and 6 (each appearing twice) beats a die with faces 3, 5, and 7, which beats a die with 4, 9, and 2, which beats 8, 1, 6. I think this may be Gardner's eample but I don't have either book handy to check. Gerry Myerson (gerry@mpce.mq.edu.au)