Mathematics · the experiment
Close the loop
Dice with faces from 1 to 6, each beating the next by a hair, the last one beating the first. Four questions, from beginner to expert, on duels won 19 times out of 36.
1 · Find the smallest number of faces
Beginner
With faces valued from 1 to n, what is the smallest number of faces for which n dice can form a loop, each beating the next by the slimmest margin?
2 · Count the duels won
Die A: 1, 3, 3, 4, 4, 4. Die B: 2, 2, 3, 3, 3, 6. Both are rolled at once: 36 equally likely pairings of faces.
Intermediate
In how many of the 36 pairings does A show the higher number?
3 · Flip the dice
In the first loop, A = (1, 1, 2, 5, 5, 5) beats B = (1, 3, 4, 4, 4, 4) in 19 duels out of 36. Replace every face v by 7 − v: A becomes A′ = (2, 2, 2, 5, 6, 6) and B becomes B′ = (3, 3, 3, 3, 4, 6).
Expert
Who wins the duel between A′ and B′?
4 · Judge the loop of fourteen
With seven faces, valued from 1 to 7, there are 49 equally likely pairings, and the slimmest win takes 25 of them. The search found a loop of 14 seven-sided dice, each beating the next by that margin.
Expert
What does this loop mean for seven-sided dice?
Loop closed
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