There are balance-scale puzzles in math. The two-pan balance scale is an important part of rural bazaars in the developing world. The simplest version consists of a metal beam that hangs two pans at the same distance.

Science and art use the double-pan balance scale. It is the basis of the concept of weight and mass in science. A two-pan scale is a symbol of balance, equality and justice in the arts and humanities.

A bi-monthly puzzle celebrating the sudden insights and unexpected twists of scientific problem solving.

The balance scale is an endless source of puzzles in which objects are balanced against each other in order to find counterfeit coins. The real coins are heavier than the fakes. These puzzles require a lot of logic and are great for math training. They teach the basics of generalization, which leads to the pursuit of formulae to describe how the number of coins you can find is related to the number of times you can weigh them. Adding all kinds of conditions to the mix will allow you to create many variations of these puzzles.

Two classics, followed by three variations with added complications are some of my favorites. Even when it isn't asked for explicitly, you can always find a general formula.

We don't give standard weights for the real coins in these puzzles. The coins need to be weighed against one another. It is assumed that the balance is sensitive enough to detect a heavy coin.

Puzzle 1

There are eight coins in this picture. The others have the same weights but one is lighter. The bad coin can be found in the weighings. The general formula for the maximum number of coins can be found here.

Puzzle 2

There are 12 coins in this picture. The others have the same weight, but one is heavier.

  1. Find the bad coin in three weighings.
  2. What is the maximum number of coins for which you can find the bad one in four weighings? Describe how you would find the fake coin.

Puzzle 3

This is a different puzzle. One of the coins is lighter than the other ones. You now have scales. Sometimes two of the scales work, but the third is broken and gives random results. You can't tell which scale is broken. It will take a lot of weighings to find the light coin.

Puzzle 4

Eight of the coins are heavy and the other eight are light. The other eight are not much different. You don't know which coins are heavier or lighter. There is one coin that has special markings. Can you tell if the coin is light or heavy on a scale? What is the maximum amount of coins you can use to solve this problem?

Balance-scale puzzles have always seemed to me to be arbitrary. How can you tell the difference between a bad coin and a good coin? I thought about that and asked the last puzzle question.

Puzzle 5

Some of the coins are fake and lighter than the others. There is at least one counterfeit coin, and there are more normal coins than counterfeit ones. It's your job to detect the fake coins.

The number of weighings increases as the number of coins decreases. It's possible that it's a new entry for the online encyclopedia of strings. Is it well known?

It's clear that we could come up with a way to test a lot of coins. If you have a favorite variant that highlights a different mathematical insight, please let us know in the comments.

Stay balanced, and be happy puzzling.

The reader who submits the most interesting, creative or insightful solution in the comments section will be given a T-shirt or one of the two Quanta books, Alice and Bob Meet the Wall of Fire. If you would like to suggest a puzzle for a future Insights column, please submit it as a comment below, marked "New Puzzle Sagittarius." The solution to the puzzle above should be submitted separately.