Problematics | The weight of medals in a sporting event

Aug 14, 2023 12:50 PM IST

Six identical medals are distinguishable by their colours, but three are heavier than the others. Can you determine which one is which with just two weighings?

It is well known that gold medals in sports aren’t really made of gold. In the 2020 Tokyo Olympics, for example, their composition was 6g of gold plating on pure silver. All medals were 85mm in diameter, but each gold medal weighed 556g, each silver medal (pure silver) was 550g, and each bronze medal (red brass) was 450g.

Welcome to Problematics!(Shutterstock) PREMIUM
Welcome to Problematics!(Shutterstock)

Not every sporting event, however, can afford medals of pure silver or a gold plating. They make do with cheaper materials, tinting them in the required colours.

#Puzzle 51.1
#Puzzle 51.1

The organisers of a small sporting event are finicky within their limited resources. They insist that gold, silver and bronze medals should all weigh the same. Two designers send their models: one gold, one silver and one bronze each. The two sets look identical; any medal from one set is indistinguishable from its counterpart in the other set.

The organisers have a scale balance but no weights. A secretary weighs the medals against one another, puts all six into her pocket, and goes to the chairman to report her findings.

Secretary: Within each set, gold = silver = bronze as required. But the medals from M/s Heavyweight & Co are lighter than those from M/s Lightweight & Sons.

Chairman: Place an order for the lighter ones, and send the heavier ones back to Lightweight & Sons.

Secretary: I can no longer identify which one came from whom. They’re in my pocket, all mixed up.

Chairman: You weighed them once against one another, didn’t you? Do that again.

Can she do it in just two weighings?

#Puzzle 51.2

The password to a safe consists of 4 digits. A safecracker has a device with four lights, which may turn green, turn yellow, or remain unlit depending on the combination entered. One green light, for example, means a digit in the right place, two yellows mean two digits in the wrong positions, and one unlit cell means that digit is not in the required combination.

The safecracker tries the following combinations one by one, getting no greens but a total of 10 yellows, before the safe’s owner catches him in the act:

#Puzzle 51.2
#Puzzle 51.2

Mailbox: Last week’s solutions

#Puzzle 50.1
#Puzzle 50.1

In this case, the revolver is rotated after each shot. So, the empty and loaded chambers are at random positions.

(a) For Target #1, the hammer can be at any chamber. So, 3 favourable and 3 unfavourable positions; survival probability = 3/6 = 1/2 (50%).

(b) As Target #1 survives, Target #2 can face any of the 2 remaining empty chambers (favourable) or 1 of the 3 loaded chambers (unfavourable). So, survival probability = 2/5 (40%).

(c) As Target #2 also survives, Target #3 can face either the sole remaining empty chamber or one of the 3 loaded chambers. So, 1 favourable and 3 unfavourable positions; survival probability = 1/4 = 25%.

— Sunita & Naresh Dhillon, Gurgaon

Solved both puzzles: Akshay Bakhai (Mumbai), Sunita & Naresh Dhillon (Gurgaon), Prof Anshul Kumar (Delhi), Shawn Jacob (Navi Mumbai), Shruti M Sethi (Ludhiana), Geetansh (Singhola, Delhi), Shivika Gupta (Delhi), Madhuri Patwardhan (Thane), Shishir Gupta (Indore)

Solved #Puzzle 50.1: Shri Ram Aggarwal (Delhi)

Solved #Puzzle 50.2: Amrish Garg (New Jersey), Dr Aishvarya Kaushik (Delhi), Dushyant Goyal (Noida), Anay Gupta (Indirapuram), Jawahar Lal Aggarwal (Vasundhara, Ghaziabad), Rozina Sehgal, Aziza

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    Puzzles Editor Kabir Firaque is the author of the weekly column Problematics. A journalist for three decades, he also writes about science and mathematics.

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