The Twelve-Coin Puzzle: Why Three Weighings Are Enough

A balance scale, twelve coins and three weighings. Here is the full answer, and why the rules you start with decide how far you can go.

You have a pile of coins and a balance scale. One coin is off. You get a few weighings to find it. That is the whole puzzle, and it has fooled smart people for generations.

The challenge: Twelve coins look identical. Eleven weigh the same. One is different, but you do not know whether it is heavier or lighter. Using a two-pan balance only three times, find the odd coin and say whether it is heavy or light.

Think you have it?

Pause here if you want to try it. Grab twelve coins, or twelve scraps of paper numbered 1 to 12. The hint is small. Do not think about which coin to test. Think about what each weighing can tell you, and plan all three before you touch the scale.

What the balance puzzle asks

A balance puzzle is a logic puzzle about finding the item that weighs differently from the rest, usually a coin, with a balance scale used a limited number of times. The scale gives you only three answers: left is heavier, right is heavier, or they are equal. So the puzzle is less about arithmetic and more about how much information you squeeze out of each weighing.

The classic forms are the nine-coin and twelve-coin problems. The twelve-coin version appeared in print as early as 1945, and it was popular on both sides of the Atlantic during the Second World War. One retelling, quoting Guy and Nowakowski, says someone suggested dropping it over Germany to sabotage their war effort. Treat that as a good story, not verified history.

Starting with the smallest case

Begin with three coins, one of them lighter. One weighing does it. Put two coins on the scale. If they balance, the coin you left off is the light one. If they do not, the lighter pan holds it. Quanta's column starts in this same place.

A brass balance scale tipped toward one side on a dark wooden table, with twelve similar-looking coins laid out in a row, one subtly different in color and shine, under warm light against a teal and amber backdrop. — Three coins, one weighing: two on the pan, one waiting on the table.

Bigger sets work the same way. Split the coins into equal groups, weigh groups against each other, and use the result to isolate the group holding the odd coin. Then repeat on the smaller group. Each extra weighing multiplies the number of coins you can handle by three. That is why eight coins with one known lighter fall in two weighings in Quanta's version, and why the limit there is 3 to the power of the number of weighings.

Answer: the three-weighing method

This is the standard solution for twelve coins, direction unknown. It is a well-known method, though the sources I drew on do not spell it out. Number the coins 1 to 12.

Weighing 1. Put coins 1 to 4 against coins 5 to 8. Coins 9 to 12 stay off.

If the pans balance, the odd coin is among 9 to 12, and coins 1 to 8 are all good. Weigh 9, 10 and 11 against 1, 2 and 3 (known good).

If they balance, coin 12 is odd. Weigh it against any good coin to learn whether it is heavy or light.
If the left pan is heavier, the odd coin is one of 9, 10, 11 and it is heavy. Weigh 9 against 10. The heavier one is odd. If they balance, it is 11.
If the left pan is lighter, the same steps apply, but the odd coin is light.

If the first weighing does not balance, say 1 to 4 is heavier. Then coins 9 to 12 are good. The odd coin is either one of 1 to 4 (heavy) or one of 5 to 8 (light). Weigh coins 1, 2 and 5 against coins 3, 6 and a good coin, say 9.

If they balance, the odd coin is 4 (heavy), 7 or 8 (light). Weigh 7 against 8. The lighter one is odd. If they balance, coin 4 is heavy.
If the left pan is heavier, the odd coin is 1 or 2 (heavy), or 6 (light). Weigh 1 against 2. The heavier one is odd. If they balance, coin 6 is light.
If the left pan is lighter, the odd coin is 3 (heavy) or 5 (light). Weigh 3 against a good coin. If it is heavy, it is coin 3. If it balances, coin 5 is light.

If the first weighing leans the other way, swap the roles of the two groups. Every branch ends in three weighings or fewer.

Choosing the right variant and formula

Before you solve anything, check the rules. Is the odd coin known to be lighter or heavier? Do you need to name the coin and its direction, or only to find a coin that is different? The answer changes the coin limit, according to the balance puzzle overview.

Direction known: up to 3^n coins in n weighings.
Direction unknown, and you must find the coin and whether it is heavy or light: up to (3^n - 3)/2 coins. For three weighings that is 12.
Direction unknown, and you only need to identify the different coin, not say whether it is heavy or light: up to (3^n - 1)/2 coins. For three weighings that is 13.

That is why twelve fits three weighings in the full version, and why a solution built for one version can fail on another. Ask three questions first. How many weighings are allowed? Is a reference coin available? Do I need only the coin, or its direction too?

NRICH offers the twelve-coin problem as a challenge for ages 14 to 16, rated level 1 of 3. It notes that three weighings are enough, and that students also found a working method using four. So a puzzle can have a minimum answer and several valid ones.

Why these puzzles reward clear reasoning

Quanta's columnist describes balance puzzles as needing clear, logical reasoning that lends itself to mathematical generalization. You build a plan, test it against every possible outcome, and then notice the rule scaling from three coins to twelve.

If you are teaching this or practicing it yourself, climb in steps. Do the three-coin case, then nine, then twelve. Say the weighing limit out loud before you start, so you know which version you are playing. The sources do not cover difficulty grading, so that part is just practical advice.