## May 28, 2019

Presentation

#### Submission Title

Nonstandard Dice That Both Count For Card Craps

#### Session Title

Session 1-3-D: Cards and Dice

#### Start Date

28-5-2019 1:45 PM

#### End Date

28-5-2019 3:10 PM

#### Disciplines

Discrete Mathematics and Combinatorics

#### Abstract

The Pala Casino in California deals Card Craps using a red die numbered {2; 2; 2; 5; 5; 5} and a blue die numbered {3; 3; 3; 4; 4; 4}. Two cards from a special 36-card deck, which contains one card bearing each of the 36 ways in which two dice can land when rolled, are dealt: one each face down to a red space and a blue space. When the dice are rolled, the higher number determines which of the cards is flipped over.

A moment's reflection reveals that Pala's blue die is unnecessary. The card selection process can be streamlined by looking only at the red die:

If the red die shows a 2, turn over the blue card.

If the red die shows a 5, turn over the red card.

While this is certainly convenient for Pala's craps dealers, this talk will determine how many ways there are to renumber the red and blue dice so that the following criteria are met:

1. Only the numbers 1-6 are used.

2. No ties are possible.

3. Each die has a 50% chance of bearing the higher number when thrown.

4. Both dice need to be consulted on at least some rolls.

#### Keywords

Nonstandard dice, Card craps

#### Author Bio

Mark Bollman is Professor of Mathematics and Chair of the Department of Mathe- matics and Computer Science at Albion College in Albion, MI. He is the author of two books on gambling mathematics: Basic Gambling Mathematics: The Numbers Behind The Neon and Mathematics of Keno and Lotteries, both published by CRC Press. Mark has taught several gambling-themed courses at Albion, which include travel to casinos in Michigan and Nevada.

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May 28th, 1:45 PM May 28th, 3:10 PM

Nonstandard Dice That Both Count For Card Craps

The Pala Casino in California deals Card Craps using a red die numbered {2; 2; 2; 5; 5; 5} and a blue die numbered {3; 3; 3; 4; 4; 4}. Two cards from a special 36-card deck, which contains one card bearing each of the 36 ways in which two dice can land when rolled, are dealt: one each face down to a red space and a blue space. When the dice are rolled, the higher number determines which of the cards is flipped over.

A moment's reflection reveals that Pala's blue die is unnecessary. The card selection process can be streamlined by looking only at the red die:

If the red die shows a 2, turn over the blue card.

If the red die shows a 5, turn over the red card.

While this is certainly convenient for Pala's craps dealers, this talk will determine how many ways there are to renumber the red and blue dice so that the following criteria are met:

1. Only the numbers 1-6 are used.

2. No ties are possible.

3. Each die has a 50% chance of bearing the higher number when thrown.

4. Both dice need to be consulted on at least some rolls.