Coins of Three Different Weights

TL;DR

This paper investigates optimal strategies for sorting coins of three known weights using balance scales with limited or unlimited capacity, providing exact bounds and analyzing special cases.

Contribution

It establishes the minimal number of weighings needed for sorting coins with a single-coin pan and extends results to unlimited capacity scenarios and special weight distributions.

Findings

Optimal weighings for single-coin pan: eil 3n/2 ceil -2

Sorting with unlimited capacity: n+1 weighings

Analysis of cases with exactly one middle-weight coin

Abstract

We discuss several coin-weighing problems in which coins are known to be of three different weights and only a balance scale can be used. We start with the task of sorting coins when the pans of the scale can fit only one coin. We prove that the optimal number of weighings for $n$ coins is $\lceil 3n/2\rceil -2$. When the pans have an unlimited capacity, we can sort the coins in $n+1$ weighings. We also discuss variations of this problem, when there is exactly one coin of the middle weight.