Chameleon Coins

TL;DR

This paper explores coin-weighing puzzles involving a chameleon coin that can mimic real or fake coins, establishing bounds and introducing solution scaling techniques for identifying fake coins among a mix of real, fake, and chameleon coins.

Contribution

It introduces bounds and a novel solution scaling method for coin-weighing problems involving chameleon coins, expanding the understanding of such puzzles.

Findings

Derived bounds for coin counts with given weighings

Introduced the concept of solution scaling

Analyzed the complexity of identifying fake coins

Abstract

We discuss coin-weighing problems with a new type of coin: a chameleon. A chameleon coin can mimic a fake or a real coin, and it can choose which coin to mimic for each weighing independently. We consider a mix of $N$ coins that include exactly two non-real coins: one fake and one chameleon. The task is to use a balance to find two coins one of which has to be fake. We find bounds for the number of coins for which we can find a solution in a given number of weighings. We also introduce an important idea of solution scaling.