The geometry of coin-weighing problems

Given a set of m coins out of a collection of coins of k unknown distinct weights, the authors wish to decide if all the m given coins have the same weight or not using the minimum possible number of weighings in a regular balance beam. Let m(n,k) denote the maximum possible number of coins for which the above problem can be solved in n weighings. They show that m(n,2)=n^(½+o(1))n, whereas for all 3⩽k⩽n+1, m(n,k) is much smaller than m(n,2) and satisfies m(n,k)=Θ(n log n/log k). The proofs have an interesting geometric flavour; and combine linear algebra techniques with geometric probabilistic and combinatorial arguments