The 12-Coin Challenge: Find a Counterfeit in Just Three Weighings
The Challenge: Twelve Coins, Three Chances
Twelve coins look identical, but one is counterfeit. It weighs either more or less than the other eleven—you do not know which. With a two-pan balance and just three weighings, can you identify the coin and tell whether it is heavy or light? Yes. The secret is to make every weighing separate the remaining possibilities, not simply compare random piles.
You need a balance scale, not a digital scale: each weighing tells you only whether the left side is heavier, the right side is heavier, or the pans balance. Number the coins 1 through 12, then keep those numbers attached to the coins as you move them.
If you want to try before reading the solution, set out twelve numbered counters and sketch the three possible results of your first weighing. The closely related twelve-ball weighing puzzle from Maths Is Fun makes a good practice version.
Why Three Weighings Can Be Enough
Each weighing has three possible outcomes. Three weighings can therefore produce up to 3 × 3 × 3 = 27 different sequences of results.
Meanwhile, there are 24 possibilities to distinguish: any one of twelve coins could be heavy, or any one could be light. That leaves enough potential result sequences—but the arithmetic alone does not give us a solution. We must choose weighings that make the useful outcomes distinct. With only two weighings, there would be at most 3 × 3 = 9 result sequences, nowhere near enough. For a deeper look at how such outcomes can be organized, see Plus Maths’ explanation of a counterfeit-weight puzzle.
That is the amazing feat here: not lifting twelve coins, but planning a tiny decision tree. After each result, you take a different path. If branching puzzles appeal to you, Puzzles Arcade’s guide to making hard puzzles easier with diagrams and tables shows why writing possibilities down can be so powerful.
Weighing One: Divide the Coins Four, Four, Four
Put 1, 2, 3, 4 on the left pan and 5, 6, 7, 8 on the right. Leave 9, 10, 11, 12 off the scale.
Now follow the result:
- Balanced: Coins 1–8 are genuine. The counterfeit is among 9–12.
- Left heavier: One of 1–4 is heavy, or one of 5–8 is light. Coins 9–12 are genuine.
- Right heavier: One of 1–4 is light, or one of 5–8 is heavy. Coins 9–12 are genuine.
Notice the value of the coins left aside. If the scale tips, they become known genuine coins that we can use as reference weights. This four-against-four start is also the foundation of mathematician Tanya Khovanova’s walkthrough of the twelve-coin problem.
If Weighing One Balances
The first eight coins are genuine, so compare 9, 10, 11 on the left with 1, 2, 3 on the right for weighing two. The right-hand coins serve as your genuine reference group.
If the Left Side Is Heavier
One of 9, 10, 11 is heavy. For weighing three, compare 9 against 10:
- Left heavier: 9 is heavy.
- Right heavier: 10 is heavy.
- Balanced: 11 is heavy.
If the Right Side Is Heavier
One of 9, 10, 11 is light. Again, compare 9 against 10:
- Left heavier: 10 is light.
- Right heavier: 9 is light.
- Balanced: 11 is light.
If They Balance Again
Coins 9–11 are genuine too, so 12 must be the counterfeit. Compare 12 against 1 for weighing three. If 12’s side goes down, it is heavy; if it goes up, it is light.
That entire branch works because a balanced first weighing provides eight trustworthy coins. But what if the scale tipped at the start?
If Weighing One Tips
Whether the first weighing went left or right, use the same setup for weighing two:
Left pan: 1, 2, 5. Right pan: 3, 6, 9.
Coin 9 is known genuine because it sat out the first, unbalanced weighing. The table below gives every possible result. Read “left heavier” and “right heavier” as the result of weighing two; the leftmost column reminds you what happened in weighing one.
| Weighing one | Weighing two | Possibilities still open | Weighing three | |---|---|---|---| | Left heavier | Left heavier | 1 heavy, 2 heavy, 6 light | Compare 1 vs. 2. The heavier coin is counterfeit; if they balance, 6 is light. | | Left heavier | Balanced | 4 heavy, 7 light, 8 light | Compare 7 vs. 8. The lighter coin is counterfeit; if they balance, 4 is heavy. | | Left heavier | Right heavier | 3 heavy, 5 light | Compare 3 vs. 9. If 3 is heavier, 3 is heavy; if they balance, 5 is light. | | Right heavier | Left heavier | 3 light, 5 heavy | Compare 5 vs. 9. If 5 is heavier, 5 is heavy; if they balance, 3 is light. | | Right heavier | Balanced | 4 light, 7 heavy, 8 heavy | Compare 7 vs. 8. The heavier coin is counterfeit; if they balance, 4 is light. | | Right heavier | Right heavier | 1 light, 2 light, 6 heavy | Compare 1 vs. 2. The lighter coin is counterfeit; if they balance, 6 is heavy. |
The table may look busy, but you only follow one row when playing. Every row ends with a third weighing that settles both questions: which coin? and heavy or light?
The Clever Move Hidden in Weighing Two
Why mix coins from opposite sides of the first weighing? Consider the first row. The first weighing says coins 1 and 2 might be heavy, while coin 6 might be light. In the second weighing, 1 and 2 sit on the left and 6 sits on the right. Either kind of counterfeit would make the left side heavier. That result groups exactly three possibilities together—just enough for the last weighing to separate.
Other candidates move differently or stay off the scale, creating other results. Rather than repeatedly weighing the same groups, the solution gives each suspect a revealing pattern of appearances. This is divide-and-conquer puzzle solving in miniature: turn one daunting mystery into a few small, manageable choices.
There is one reassuring detail in the table. In the two-possibility rows, one apparent result of the final weighing cannot occur under the puzzle’s rules. For example, if the first weighing was left-heavy and the second right-heavy, comparing 3 against genuine coin 9 can show 3 heavy or can balance, revealing 5 light. Coin 3 cannot come out lighter in that branch. An “impossible” outcome is not a missing answer; it is a check that something went wrong while setting up or recording the weighings.
Try the Feat for Yourself
This puzzle becomes much friendlier when you act it out. Ask someone to choose a numbered counter secretly and label it “heavy” or “light.” They can then announce what a balance scale would show at each step while you follow the appropriate branch. Swap roles and try again with a different counterfeit.
The lesson goes beyond coins. A useful question is not merely “What can I test next?” but “What will each possible result tell me?” Three carefully planned weighings answer a question that dozens of unplanned comparisons might only muddle.
