Balance-scale puzzles abound in recreational mathematics. The essential element is the humble two-pan balance scale — a staple of commerce over the millennia that’s still found in bustling rural bazaars in the developing world. The simplest versions consist of a metal beam from which hang two pans at equal distances from the central support or fulcrum.

In recreational mathematics, the balance scale is an endless source of puzzles in which objects, generally coins, are balanced against each other in order to find the counterfeit coins among them. The counterfeit coins are either heavier or lighter than the real ones. These puzzles are excellent math training tools — they require precise and elaborate logic requiring all eventualities to be thought out in detail. In addition, they teach the fundamentals of generalization, leading naturally to the pursuit of formulae to describe how the number of coins you can successfully search changes in relation to the number of times you’re allowed to weigh the coins. And finally, you can create countless variations of these puzzles by adding all kinds of conditions to the mix.

Here are some of my favorite such problems, starting with two classics, followed by three variations with added complications. In each case, you can always look for a general formula, even when it is not asked for explicitly.

Note that in all these puzzles, we do not provide standard weights for the real coins. You have to weigh the coins against each other. Also, it is assumed that the balance is sensitive enough to detect a single light or heavy coin among the standard ones.

Clearly, we could come up with ways to test an endless combination of coins. If you have a favorite variant that highlights a different mathematical insight, please feel free to post it in the comments section below.

Happy puzzling, and stay balanced!