Abstract
Suppose we can and do place k distinct symbols a 1, a 2, …, a k on the vertices of n − 1 directed n-gons in such a way that each symbol appears exactly once on each n-gon, and each pair of symbols occurs once in some n-gon at each of the n − 1 possible directed distances. Then for k = 3 and k = 4, as above, and for each circular order ( πa 1, πa 2,…, πa k ), the number of n-gons with the symbols in that order clockwise must be equal to the number of n-gons with the symbols in that order counterclockwise. A counterexample found by Tuvi Etzion shows that when k = 5, the conclusion does not necessarily follow.
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