# The 5-Way Scale

## TL;DR

This paper explores coin-weighing problems using a 5-way scale, demonstrating that it can identify two fake coins among 2^k coins in fewer weighings than a 3-way scale.

## Contribution

It introduces a novel strategy for 5-way scales that outperforms 3-way scale strategies in finding two counterfeit coins efficiently.

## Key findings

- 5-way scale provides more information per weighing
- Strategy finds two fake coins among 2^k coins in k+1 weighings
- 5-way scale strategy is more efficient than 3-way scale methods

## Abstract

In this paper, we discuss coin-weighing problems that use a 5-way scale which has five different possible outcomes: MUCH LESS, LESS, EQUAL, MORE, and MUCH MORE. The 5-way scale provides more information than the regular 3-way scale. We study the problem of finding two fake coins from a pile of identically looking coins in a minimal number of weighings using a 5-way scale. We discuss similarities and differences between the 5-way and 3-way scale. We introduce a strategy for a 5-way scale that can find both counterfeit coins among $2^k$ coins in $k+1$ weighings, which is better than any strategy for a 3-way scale.