Having young kids, we seem to have a lot of games around the house at the moment. The geek in me starts looking at them as maths challenges. Beetle, by Milton Bradley, is a variant on the classic dice game. I was curious to use Markov chains to figure out the average game length, and which positions were closer to victory.

Basic rules:

- Take turns spinning the spinner to build a piece of the beetle
- 1/6 chance of getting the body
- 1/6 chance of getting the head
- 1/6 chance of getting one of two eyes
- 1/6 chance of getting one of two antenna
- 2/6 chance of getting one of six legs
- Must get the body before getting anything else
- Must get the head before the antenna or eyes

This allows for 71 states as depicted in the diagram below:

The 71x71 matrix is basically formed from the above diagram. There are three regions:

- no body
- body, but no head
- body and head

Within each region there are various combinations of legs or other pieces, as permitted. E.g. state 'A' has no pieces; state 'B' has a body, no head, and five legs; state C has a head and body, one antenna, two eyes and one leg; state 'D' is a completed beetle. Each successful spin moves either to the right or down (or 'out') to an adjacent state.

Result: **31.498** spins on average.

The following table shows the average number of moves remaining to victory from each position, sorted by moves remaining (hopefully the notation is self explanatory):

Combination | Average turns to victory |
---|---|

b h e2 a2 l6 | 0.00000 |

b h e2 a2 l5 | 3.00000 |

b h e1 a2 l6 | 6.00000 |

b h e2 a1 l6 | 6.00000 |

b h e2 a2 l4 | 6.00000 |

b h e1 a2 l5 | 7.00000 |

b h e2 a1 l5 | 7.00000 |

b h e1 a2 l4 | 8.66667 |

b h e2 a1 l4 | 8.66667 |

b h e1 a1 l6 | 9.00000 |

b h e2 a2 l3 | 9.00000 |

b h e1 a1 l5 | 9.50000 |

b h e1 a1 l4 | 10.58333 |

b h e1 a2 l3 | 10.77778 |

b h e2 a1 l3 | 10.77778 |

b h a2 l6 | 12.00000 |

b h e2 l6 | 12.00000 |

b h e2 a2 l2 | 12.00000 |

b h e1 a1 l3 | 12.18056 |

b h a2 l5 | 12.33333 |

b h e2 l5 | 12.33333 |

b h a2 l4 | 13.11111 |

b h e2 l4 | 13.11111 |

b h e1 a2 l2 | 13.18519 |

b h e2 a1 l2 | 13.18519 |

b h a1 l6 | 13.50000 |

b h e1 l6 | 13.50000 |

b h a1 l5 | 13.70833 |

b h e1 l5 | 13.70833 |

b h e1 a1 l2 | 14.18287 |

b h a1 l4 | 14.27778 |

b h e1 l4 | 14.27778 |

b h a2 l3 | 14.33333 |

b h e2 l3 | 14.33333 |

b h e2 a2 l1 | 15.00000 |

b h a1 l3 | 15.26736 |

b h e1 l3 | 15.26736 |

b h e1 a2 l1 | 15.79012 |

b h e2 a1 l1 | 15.79012 |

b h a2 l2 | 15.95062 |

b h e2 l2 | 15.95062 |

b h e1 a1 l1 | 16.48650 |

b h l6 | 16.50000 |

b h l5 | 16.60417 |

b h a1 l2 | 16.66705 |

b h e1 l2 | 16.66705 |

b h l4 | 16.94097 |

b h l3 | 17.60417 |

b h a2 l1 | 17.89712 |

b h e2 l1 | 17.89712 |

b h e2 a2 | 18.00000 |

b h a1 l1 | 18.42943 |

b h e1 l1 | 18.42943 |

b h e1 a2 | 18.52675 |

b h e2 a1 | 18.52675 |

b h l2 | 18.63561 |

b h e1 a1 | 19.00662 |

b h l1 | 20.03252 |

b h a2 | 20.10700 |

b h e2 | 20.10700 |

b h a1 | 20.49312 |

b h e1 | 20.49312 |

b h | 21.76282 |

b l6 | 22.50000 |

b l5 | 22.53472 |

b l4 | 22.67014 |

b l3 | 22.98148 |

b l2 | 23.53286 |

b l1 | 24.36608 |

b | 25.49833 |

empty | 31.49833 |

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