Confirming the Labels of Coins in One Weighing

TL;DR

This paper investigates the problem of verifying labels on identical-looking coin bags with different weights using only one weighing, focusing on downhill weighings and optimizing total weight and coin count.

Contribution

It introduces the concept of downhill weighings, analyzes their importance, and determines minimal total weight and coin counts needed for label confirmation.

Findings

Identifies the smallest total weight for downhill weighings that confirm labels.

Establishes bounds on the minimum number of coins required for such weighings.

Highlights the significance of downhill weighings in coin label verification.

Abstract

There are $n$ bags with coins that look the same. Each bag has an infinite number of coins and all coins in the same bag weigh the same amount. Coins in different bags weigh 1, 2, 3, and so on to $n$ grams exactly. There is a unique label from the set 1 through $n$ attached to each bag that is supposed to correspond to the weight of the coins in that bag. The task is to confirm all the labels by using a balance scale once. We study weighings that we call downhill: they use the numbers of coins from the bags that are in a decreasing order. We show the importance of such weighings. We find the smallest possible total weight of coins in a downhill weighing that confirms the labels on the bags. We also find bounds on the smallest number of coins needed for such a weighing.