Discreet Coin Weighings and the Sorting Strategy

TL;DR

This paper analyzes a coin weighing strategy called the sorting strategy, exploring when it remains discreet and how many fake coins can be consistent with its outcomes, connecting it to the Frobenius coin problem.

Contribution

It introduces and studies the discreetness of the sorting strategy, characterizing the possible fake coin counts and linking the problem to the Frobenius coin problem.

Findings

The number of fake coins can form an arithmetic progression.

Discreetness depends on the length of certain arithmetic subsequences.

The revealing factor for the sorting strategy is calculated.

Abstract

In 2007, Alexander Shapovalov posed an old twist on the classical coin weighing problem by asking for strategies that manage to conceal the identities of specific coins while providing general information on the number of fake coins. In 2015, Diaco and Khovanova studied various cases of these "discreet strategies" and introduced the revealing factor, a measure of the information that is revealed. In this paper we discuss a natural coin weighing strategy which we call the sorting strategy: divide the coins into equal piles and sort them by weight. We study the instances when the strategy is discreet, and given an outcome of the sorting strategy, the possible number of fake coins. We prove that in many cases, the number of fake coins can be any value in an arithmetic progression whose length depends linearly on the number of coins in each pile. We also show the strategy can be discreet when the number of fake coins is any value within an arithmetic subsequence whose length also depends linearly on the number of coins in each pile. We arrive at these results by connecting our work to the classic Frobenius coin problem. In addition, we calculate the revealing factor for the sorting strategy.