You are here

Frank Morgan's Math Chat - How to Play Fair With Loaded Dice


September 7, 2000

Questionable Mathematics. The Christian Science Monitor of July 18 reported that "More than 650 million" Americans took to the skies last year. That's more than twice the entire population! They probably meant 650 million passenger flights, counting some people many times.

Math Chat invites readers to submit further examples of questionable mathematics.

Old Challenge (Al Zimmermann). For some reason you need a standard fair die (with the numbers from 1 to 6 on the six faces). All you have is an unfair "loaded" die. How can you use the loaded die to fairly choose random numbers from 1 to 6?

Answer (José Cañizo and Arthur Pasternak). Roll the die three times. If you don't get three different numbers, start over. When you get three different numbers, the highest roll H, the middle roll M, and the lowest roll L can occur in six, equally likely patterns: HML, HLM, MHL, MLH, LHM, and LMH, which can be associated with the numbers from 1 to 6.

New Challenge. In the September ACBL Bridge Bulletin, Noreen Wurdemann of the Bahamas reports that the two bridge hands below were dealt out at her club. She says that she is "100% certain there was no hanky-panky." What is the probability of this happening among say 28 boards? somewhere in the world in a year? in a century? How do you explain this report?



Send answers, comments, and new questions by email to, to be eligible for Flatland and other book awards. Winning answers will appear in the next Math Chat. Math Chat appears on the first and third Thursdays of each month. Prof. Morgan's homepage is at

THE MATH CHAT BOOK, including a $1000 Math Chat Book QUEST, questions and answers, and a list of past challenge winners, is now available from the MAA (800-331-1622).

Copyright 2000, Frank Morgan.