Kevin Buzzard posted this to sci.math.research, and I am taking the liberty of reposting it here: << The following came up in an undergraduate problem-solving group: given 6 distinct points in the plane we can consider the 6-choose-2 distances between pairs of distinct points. If M is the largest of these distances, and m is the smallest, then show that M/m>=sqrt(3). In fact sqrt(3) isn't optimal, but I don't know what the optimal number is. One might ask what the answer is with n>=3 points in the plane: a compactness argument shows that there will be some r(n)>=1 such that M/m is always at least r(n) and furthermore such that r(n) is attained by some configuration of n points. . . . . . .
(I think I have an asymptotic formula for the optimal ratio as n -> oo, but so far not a proof. I'm curious if anyone else comes up with a conjectural asymptotic formula that is asymptotic to mine. Putative asymptotic formula is way, way below.) --Dan Guess: r(n) ~ sqrt(3n/2) as n -> oo. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele