Markov analysis of Beetle

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