February 9, 2021

Change of Face

The serious gamblers in casinos hang out at the craps tables. The basic rules of this two-dice game are simple, but the bewildering array of options for betting on various outcomes creates a fast-paced, insidiously seductive pastime, in which a heady brew of chance, intuition, experience, calculation, and superstition come into play.

The shooter tosses two dice. If a total of seven or eleven comes up on a beginning roll, the shooter and those wagering with him win whatever amount they bet. If a two, three, or twelve total (called craps) shows up, the shooter and his companions lose.

Players betting against the shooter win if a two or three comes up. They neither lose nor win for a double six (twelve). Any of the remaining six totals (four, five six, eight, nine, and ten) on a beginning roll starts off a different sequence of play, with different possible bets.

Suppose a shooter replaces the standard dice with a pair of new dice whose faces are marked as follows: 1, 2, 2, 3, 3, 4 and 1, 3, 4, 5, 6, 8. Should the other players or the casino management object?

First, we can check whether the probabilities of the various sums with the new dice are different. We can do this by displaying in a table all the ways in which each sum from two to twelve can be obtained with the two different pairs of dice.

Substituting a pair of "weird" dice with faces labeled 1, 2, 2, 3, 3, 4 and 1, 3, 4, 5, 6, 8 for a pair of standard dice has no effect on games in which only the sums matter. For both sets of dice, there's only one way to roll a sum of two, two ways to roll a sum of three, and so on, as shown in charts of sums corresponding to the possible outcomes for standard dice (left) and weird dice (right). In games in which rolling doubles plays a role, however, there is a difference. There are six ways to roll doubles with standard dice and only four ways to roll doubles with weird dice (shaded squares).

Interestingly, the unusually marked dice (known as Sicherman dice) and the standard dice have exactly the same frequencies for the possible sums. There's only one way to roll a two, two ways to roll a three, and so on.

Although the numbering on the weird dice is completely different from that on the standard dice, all the odds are exactly the same for rolling any given sum. You could make the substitution at the craps table as far as the sums are concerned (see "Weird Dice" by Joseph Gallian).

The trouble is that craps betting puts certain combinations, such as doubles, in special roles (for example, the double six on a beginning roll). With the weird dice, there's no way to get a double two, five, or six, and there are two ways to get a double three.

Changing the dice would have a considerable impact on many board games in which rolling doubles or other special combinations affects a player's fate (see "Weird Dice").

In Monopoly, players buy, sell, rent, and trade real estate in cutthroat competition to bankrupt their opponents, They take turns throwing a pair of dice , with the totals indicating how many spaces to proceed along an outside track that includes 22 color-coded properties, four railroads, two utilities, a Luxury Tax square, an Income Tax square, three Chance squares, and three Community Chest squares. Corner squares are marked Go, Just Visiting/In Jail, Free Parking, and Go to Jail.

Players start at Go. A double warrants a second throw, but three consecutive doubles sends a player directly to the In Jail square. To get out of jail, the player must throw a double. If she succeeds, whatever sum she gets decides how many spaces to advance along the board (see also "Monopoly Cheat Sheet").

However, using the nonstandard dice gives a lower probability of rolling doubles (only 4 out of 36 instead of 6 out of 36). Moreover, the chances of landing on a square six spaces away goes up twofold and the chances of landing four, ten, or twelve spaces away on a move out of jail are zero. 

Thus, if you happen to own the property St. James Place, which is six spaces away from jail, you are likely to collect lots of rent from players escaping jail. On the other hand, the owner of Virginia Avenue (four spaces away from jail) loses out on the extra business.

In fact, according to calculations made in one study, the change in dice moves St. James Place up from tenth to the sixth most frequently visited space in the game. Virginia Avenue descends from 24th to  27th in the rankings.

It's possible to prove mathematically that the weird dice represent the only alternative numbering of a pair that provides the same sum probabilities as standard dice. Of course, it doesn't matter how you arrange the six numbers on each die, though you can opt for symmetry by placing the numbers so that each set of opposite sides totals five or nine.

You can work out alternative numbering schemes not only for cubic dice but also for dice in the shape of a tetrahedron (four triangular faces), an octahedron (12 pentagonal faces), and an icosahedron (20 triangular faces). (See "Renumbering the Faces of Dice" by Duane M. Broline.)

For example, a pair of standard octahedrons marked 1, 2, 3, 4, 5, 6, 7, and 8 have the same sum probabilities as a pair with one marked 1, 3, 5, 5, 7, 7, 9, 11 and the other marked 1, 2, 2, 3, 3, 4, 4, 5. Either set would work in a two-dice game.

There are also other ways to get the same sum results as two standard cubic dice. For example, you can use a tetrahedral die labeled 1, 1, 4, 4 combined with a spinner with 18 equal segments labelled 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, and 8. This combination yields the same sum probabilities as an ordinary pair of cubes labeled 1 through 6.

Using weird dice brings a fresh perspective to games of chance. It provides a welcome opportunity for examining the interaction between rules and randomizers.

Next: The Long Run

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