Summary: Coins with arbitrary weights
Dmitry N. Kozlov
Given a set of m coins out of a collection of coins of k unknown distinct weights, we 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. It is known that m(n, 2) = n( 1
Here we determine the asymptotic behaviour of m(n, k) for larger values of k. Surprisingly
it turns out that for all 3 k n + 1, m(n, k) is much smaller than m(n, 2) and satisfies
m(n, k) = (n log n/ log k).
Coin-weighing problems deal with the determination or estimation of the minimum possible number
of weighings in a regular balance beam that enable one to find the required information about
the weights of the coins. There are numerous questions of this type, see, e.g., [GN] and its many
references. Here we study the following variant of the old puzzles, which we call the all equal problem.
Given a set of m coins, we wish to decide if all of them have the same weight or not, when various
conditions about the weights are known in advance. The case in which the coins are given out of