You have twelve coins. Eleven are identical. One is off, and you do not know whether it is heavy or light. You get a balance scale and three weighings. That is the whole puzzle, and it is harder than it sounds.
Counting Outcomes, Not Pounds
The scale never tells you a weight here. It gives one of three answers: left side down, right side down, or level. So each weighing yields at most three outcomes, and k weighings yield at most 3^k distinct results.
Now count what you need to tell apart. If you know the odd unit is heavy, each unit is one scenario. If you do not know the direction, each unit is two scenarios: heavy or light. That doubles the demand. Three weighings give 27 outcomes. Twelve units with two directions each make 24 scenarios, which fits. Thirteen units would make 26, and the standard result is that three weighings can still find the odd one among thirteen, but not always say whether it is heavy or light. In general, k weighings handle (3^k - 1)/2 units when you only need to find the odd one. Thirteen is that number for k = 3.
A Worked Solution for Twelve
Number the units 1 to 12. First, weigh 1 to 4 against 5 to 8. Everything after depends on the result.
If the scale balances, units 1 to 8 are genuine. The odd unit is among 9 to 12. Weigh 9, 10 and 11 against 1, 2 and 3. If that balances, unit 12 is odd, and one more weighing against any genuine unit shows heavy or light. If it tips, you now know the direction, because the genuine units sit on one side. Say 9 to 11 went down. Then the odd one is heavy and among those three. Weigh 9 against 10. The lower side is the culprit. If level, it is 11.
If the first weighing tips, say left down, the direction is still open. Either one of units 1 to 4 is heavy, or one of 5 to 8 is light. Units 9 to 12 are genuine. This is where the earlier shortcut fails. A tilt does not point to a direction. It points to a pool of suspects with two possible stories. Weigh 1, 2 and 5 against 3, 6 and 9, where 9 is a known genuine filler.
- Level: unit 4 is heavy, or unit 7 or 8 is light. Weigh 7 against 8. The lighter one is odd. If level, it is 4.
- Left down: unit 1 or 2 is heavy, or unit 6 is light. Weigh 1 against 2. The heavier one is odd. If level, it is 6.
- Right down: unit 3 is heavy, or unit 5 is light. Weigh 3 against a genuine unit to settle it.
Three weighings, every branch closed. The mirror case, right down on the first weighing, works the same way with the sides swapped.
Where People Go Wrong
The common mistake is to treat a tilt as if it names the heavy side. With an unknown direction, left down means a heavy unit on the left or a light unit on the right. Drop one of those and the plan breaks.
Another mistake is splitting into halves. A half-and-half split throws away the third outcome. Thirds use all three. Keeping a ledger helps. Write each suspect as a pair, such as 3H or 6L, and strike names off as results arrive. Genuine units are also worth tracking, because they become the filler you need later. Unit 9 in the example does exactly that job.
Why Real Inspectors Sample Instead
The puzzle promises exactly one bad unit. Real packaging does not. NIST Handbook 133 is a procedural guide for checking that the net contents of packaged goods match what the label says [1]. It was written mainly for state, local and some federal officials, though NIST says packagers and sellers may find it useful too.
The question there is about a lot, not a single suspect. A NIST training seminar on the handbook's basic material listed sampling, quantity control and calculating standard deviation among its topics [2]. So the working method is to check a subset and judge conformance from it. The sources describe the approach but not why it was chosen, so cost and practicality are a reasonable guess, not a documented reason.
What the Sources Do Not Say
The sources do not cover the balance puzzle at all. The weighing strategy above is standard recreational logic, not something drawn from NIST.
They also give no sample sizes, tolerance values or acceptance thresholds, and the excerpt of the 2026 edition cuts off before its details. Nor do they explain how standard deviation is used to judge a lot. Readers who need those numbers should go to the handbook itself.
The contrast is still useful. The puzzle gives certainty from a guaranteed single flaw and a fixed number of weighings. Sampling gives a judgment about a whole lot. Different problems, different tools.
Sources
- NIST Handbook 133: Checking the Net Contents of Packaged Goods (2026 Ed.) | NIST
- Handbook 133 - Checking the Net Contents of Packaged Goods - Basic | NIST
- https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQEnZ3bAULdgusXhdhkILcrzd2i9P2NT5EZPxaoyu3K9y7xoBIZLvmZgW9U3eiUnc1ZKvIrtL13auWqxmYDegkK304Xn7ch5YygvLoq83OXF6pCEAeEqdYZ2nfnZ0SnLBEf5-PAxOB4=
Siobhan O'Reilly investigates the friction between urban decay and digital expansion for Deeperpectives. She spent a decade reporting from transit hubs and temporary settlements across the rust belt. Her work prioritizes the raw data of lived experience over polished corporate narratives.
