## Project Euler 205: Comparing four-sided and six-sided dice

#### Project Euler 205 Problem Description

Project Euler 205: Peter has nine four-sided (pyramidal) dice, each with faces numbered 1, 2, 3, 4.

Colin has six six-sided (cubic) dice, each with faces numbered 1, 2, 3, 4, 5, 6.

Peter and Colin roll their dice and compare totals: the highest total wins. The result is a draw if the totals are equal.

What is the probability that Pyramidal Pete beats Cubic Colin? Give your answer rounded to seven decimal places in the form 0.abcdefg

#### Analysis

Type | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

9 four-sided dice | 0 | 0 | 0 | 1 | 9 | 45 | 165 | 486 | 1206 | 2598 | 4950 | 8451 | 13051 | 18351 | 23607 | 27876 | 30276 |

6 six-sided dice | 1 | 6 | 21 | 56 | 126 | 252 | 456 | 756 | 1161 | 1666 | 2247 | 2856 | 3431 | 3906 | 4221 | 4332 | 4221 |

Type | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

9 four-sided dice | 30276 | 27876 | 23607 | 18351 | 13051 | 8451 | 4950 | 2598 | 1206 | 486 | 165 | 45 | 9 | 1 |

6 six-sided dice | 3906 | 3431 | 2856 | 2247 | 1666 | 1161 | 756 | 456 | 252 | 126 | 56 | 21 | 6 | 1 |

After calculating the table above all that remains is finding the ratio of winning combinations to the total possible combinations. So the denominator will be (4^9)(6^6) and the numerator the total number of ways Pete out sums Colin (Pete’s index one more than Colin’s). Draws are inherently ignored, which was fortuitous.

#### Project Euler 205 Solution

Runs < 0.002 seconds in Python 2.7.Use this link to get the Project Euler 205 Solution Python 2.7 source.

#### Answer

Slowly swipe from either end beginning with the white vertical bar to get an idea of the starting or ending digits. For less drama, just double click the answer area. The distance between the two bars will give you an idea of the magnitude. Touch devices can tap and hold the center of the box between the two bars and choose*define*to reveal the answer.

#### Afterthoughts

- Reference: The On-Line Encyclopedia of Integer Sequences (OEIS) A108907: Number of times a point sum n is attained in all 6^6 permutations of throwing 6 dice.

Project Euler 205

*Project Euler 205 Solution last updated*

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