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Can I ask a dumb question? I understand the permutation argument for why using 5 six-sided dice can't possibly be fair, but looking at the serpentine example, I'm having trouble intuiting why. At a glance, removing any one of those dice looks pretty fair.

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If you look at just the first 5 dice and compute the probability that player one rolls lowest, of the 6^4 combinations of rolls the other players can make: 1 beats all 6^4; 16 beats 4^4, 17 beats 4^4, 32 beats 2^4, 33 beats 2^4, and 48 beats nothing. So player 1 is lowest 1840 out of 6^5, or ~23.7%.

A similar argument shows player 5 rolls lowest 1414 out of 6^5, or only 18.2% of the time.

I don't have a good intuitive explanation though.

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