Ordering 15 marbles with a three-way scale. Issue 542 (July 2014)

• Record Type: Journal Article
• Authors: Audet, Charles
• Is Part Of: Mathematical gazette. Volume 98:Issue 542(2014)
• Page Start: 304
• Page End: 316
• Publication Date: 2014-07
• Subjects: Mathematics -- Periodicals; Mathématique; Mathematik; Mathematics; Periodicals; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); Zeitschrift; Online-Publikation; 510
• DOI: 10.1017/S0025557200001340
• Languages: English
• ISSNs: 0025-5572

Abstract

Abstract : Many mathematical puzzles involve using a scale to compare the weights of two objects, or two groups of objects. The present work studies a similar but different type of question involving a scale. We are given a set of marbles, whose weights are all different. The only way to distinguish them is to use a special kind of scale. The scale has three trays and each can accept exactly one marble. The scale then indicates which is the heaviest, the lightest and consequently the middle one of the three marbles. The paper studies the question of ordering 15 different marbles with as few weighings as possible. The problem is taken from the website Enigmes, casse-têtes, curiosités et autres bizarreries [1] and Toppuzzle [2] where the best reported strategy requires 23 weighings. The problem can also be found at the website Trick of Mind [3] where a strategy requiring 22 weighings is proposed. These websites contain many challenging mathematical puzzles, some are classical problems, others are original creations. The author is not aware of any other work on this problem. The present paper describes a strategy that improves on the value of 22 weighings, and is structured as follows. Section 2 introduces some useful definitions and provides strategies to sort up to 9 marbles. Section 3 proves our main result, that 20 weighings is always sufficient to sort 15 marbles.