Number of Partitions of an n-kilogram Stone into Minimum Number of Weights to Weigh All Integral Weights from 1 to n kg(s) on a Two-pan Balance

TL;DR

This paper determines the number of partitions of an n-kilogram stone into the minimum parts needed to weigh all integral weights from 1 to n kg on a two-pan balance, with implications for optimal weight and currency design.

Contribution

It introduces a novel approach to partitioning stones for minimal parts to achieve universal weighing, considering weight constraints and practical applications.

Findings

Derived formulas for the number of such partitions.

Compared traditional and constrained partitions highlighting advantages.

Potential applications in designing optimal weights and currency denominations.

Abstract

We find out the number of different partitions of an n-kilogram stone into the minimum number of parts so that all integral weights from 1 to n kilograms can be weighed in one weighing using the parts of any of the partitions on a two-pan balance. In comparison to the traditional partitions, these partitions have advantage where there is a constraint on total weight of a set and the number of parts in the partition. They may have uses in determining the optimal size and number of weights and denominations of notes and coins.