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by CurtMonash
4594 days ago
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Quite so. I'd say there simple are 8 possible states -- nominally 12, but 2 each are collapsed via chutes and ladders. If I had to solve it by hand, however, I wouldn't try to invert an 8x8 matrix. Rather, I'd write it down as a simple system of linear equations. Let f1, ..., f12 be the expected number of steps to completion if you start in position 1, ..., 12. f12 = 0.
f11 = 1.
f10 = (f11 + f12 + f12)/3 + 1.
f9 = f8, because of the chute.
f8 = (f9 + f10 + f11)/3 + 1. and so on. |
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