Ivars Peterson's MathTrek

October 10, 2005

Binary Dice

"Iacta alea est!"—Julius Caesar (100–44 B.C.) Dice have been around for thousands of years. In 49 B.C., when Julius Caesar ordered his troops across the river Rubicon to wage civil war in Italy, the alea of the well-known proverb that he quoted already had the standard form of the die we use today: a cube engraved or painted with one to six dots, arranged so that the number of dots on opposite faces totals seven and the faces marked with one, two, and three dots go counterclockwise around a corner. Even earlier, cubic dice were used for games and gambling in classical Greece and in Iron Age settlements across northern Europe.

Dice don't have to be cubic. More than 4,000 years ago, the nobility of the Sumerian city of Ur played with tetrahedral dice. Carefully crafted from ivory or lapis lazuli, each die was marked on two of its four corners, and players presumably counted how many marked or unmarked tips faced upwards when these dice were tossed. Egyptian tombs have yielded four-sided pencils of ivory and bone, which could be flung down or rolled to see which side faces uppermost.

The simplest type of die, such as a coin, has only two sides. Such binary dice date back to the very earliest days of gaming.

Puzzle designer and expert solver Wei-Hwa Huang has come up with an intriguing set of dice that combines the cubic form of standard dice with the simplicity of binary choices. These dice lend themselves to a variety of entertaining puzzles and games. He calls them "Boolean dice." The term "binary dice" was already taken.

Boolean dice: One black die is blank, and the remaining dice in the set each have a certain number and pattern of white dots, one dot per face. Each die is unique.
Credit: Courtesy of Wei-Hwa Huang.

You can easily make a set for yourself by affixing stickers to the appropriate faces of 10 cubes.

Here's one puzzle, suggested by Huang, involving these Boolean dice.

Remove two dice. Using the other eight dice, make a two-by-two block that has a checkerboard pattern on each of its six faces. Can this be done no matter which two dice you remove?

Here's a two-player, pencil-and-paper game involving Boolean dice.

On a sheet of paper, write down two columns of numbers, from 0 through 5, one column for each player. At the top of each column, draw three little circles, which will represent "nudges" (explained below).

The two players split the 10 dice into two groups of five dice each. The object of the game is to roll five dice to get 0, 1, 2, 3, 4, and 5 white dots facing up before your opponent does. Players take turns simultaneously.

Roll all your dice. You may then cross out the number that corresponds to the number of dice with a white dot facing up. However, after your roll but before crossing off a number, you can opt to use a "nudge." Cross off one of the three little circles, then turn one die 90 degrees so that a different face is on top. Getting the bottom to face up requires two nudges. You are free to use all three nudges in one turn if you wish, but this also exhausts your supply.

After all the nudging and crossing off has happened for both players, if the number of white-up dice of one player at any time during the turn had matched the number of white-up dice of the other player, the two players must switch dice sets before the next turn. Note that you can use nudges to force a switch, but you can't use them to avoid a switch.

Whoever crosses off all their numbers first wins. If both players finish crossing off their numbers on the same turn, whoever has the most nudges left wins. If they have the same number of nudges left, the game is a draw.

Huang provides a sampling of three puzzles and six games using Boolean dice at http://www.ofb.net/~whuang/gp/booleandice/ and http://www.ofb.net/~whuang/gp/booleandice/g4g6.html. He's interested in hearing from anyone who comes up with any novel ideas about what you can do with these dice.

References:

Wei-Hwa Huang describes his "Boolean dice" at http://www.ofb.net/~whuang/gp/booleandice/. He can be reached at onigame@gmail.com.

Peterson, I. 2002. Tricky dice revisited. MAA Online (April 15).

______. 1999. Averting instant insanity. MAA Online (Aug. 9).

______. 1998. Unfair dice. MAA Online (Oct. 26).

______. 1998. The Jungles of Randomness: A Mathematical Safari. New York: Wiley.