Coins that Change Their Weights

TL;DR

This paper investigates strategies to identify a fake coin that can change its weight among multiple states using the fewest weighings on a balance scale, considering different state-change scenarios and starting conditions.

Contribution

It introduces optimal oblivious strategies for detecting a weight-changing fake coin with multiple states, expanding on previous work by analyzing more complex state transitions.

Findings

Optimal strategies for 2-state coins changing from lighter to heavier.

Analysis of 3-state coins with cyclic weight changes.

Comparison of adaptive and oblivious strategies under unknown starting states.

Abstract

As in many coin puzzles, we have several identical-looking coins, with one of them fake and the rest real. The real coins weigh the same. Our fake coin is special in that it can change its weight. The coin can pretend to be a real coin, a fake coin that is lighter than a real one, and a fake coin that is heavier than a real one. In addition to this, each time the coin is on the scale, it changes its weight in a predetermined fashion. In this paper, we seek to find our fake coin using a balance scale and the smallest number of weighings. We consider different possibilities for the fake coin. We discuss coins that change weight between two states or between three states. The 2- state coin that changes weight from lighter to real and back has been studied before, so we concentrate on the 2-state coin that changes weight from lighter to heavier, and back. We also study the 3-state coin, which changes its weight from lighter to heavier to real, and back to lighter. Given the total number of coins and the starting state of the fake coin, we calculate the smallest number of weighings needed to identify the fake coin. We provide an oblivious optimal strategy for this number of weighings. We also discuss what happens if the starting state is not known or mixed. In such cases, adaptive strategies are often more powerful than oblivious ones.