Motivated by questions in theoretical distributed computing, we develop the combinatorial theory of abstract simplex path subdivisions. Our main application is a short and structural proof of a theorem of Castañeda and Rajsbaum. This theorem in turn implies the solvability of the weak symmetry breaking task in the immediate snapshot wait-free model in the case when the number of processes is not a power of a prime number.

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/sup ( 1/2 +o(1))n/, whereas for all 3/spl les/k/spl les/n+1, m(n,k) is much smaller than m(n,2) and satisfies m(n,k)=/spl Theta/(n log n/log k). The proofs have an interesting geometric flavour; and combine linear algebra techniques with geometric probabilistic and combinatorial arguments.