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I'm still trying to wrap my head around this. Non-transitive dice are four dice and a game to go with them, where each die beats the next in line, and the last beats the first. Each can be shown to be better than the next, but somehow it keeps going in a cycle, never reaching an all-around best die. Kind of like a quantitative rock-paper-scissors, reminiscent of Escher's Ascending and Descending.
Wikipedia has more on these dice.