Every genuine token weighs exactly 10 grams. Every counterfeit token weighs exactly 11 grams.
Exactly one sack is entirely counterfeit. The other nine sacks are entirely genuine.
You have a digital scale that shows the exact weight of whatever you put on it, and you may use it only once.
How can you identify the counterfeit sack?
💡 Hint
One reading is just one number, so make that number carry more information. Ask how every source could join the weighing in a way that leaves its own fingerprint.
Reveal answer
Take 1 token from sack 1, 2 from sack 2, and so on up to 10 from sack 10. The number of grams over 550 is the number of the fake sack.
Step-by-step solution
- Number the sacks 1 to 10. Take 1 token from sack 1, 2 tokens from sack 2, and continue up to 10 tokens from sack 10. That is 1 + 2 + … + 10 = 55 tokens.
- If every token were genuine, the pile would weigh 55 × 10 = 550 grams.
- Each counterfeit token adds exactly 1 gram. If sack k is the fake one, the pile holds k fake tokens, so the scale reads 550 + k grams.
- Read the scale and subtract 550. A reading of 557 grams means sack 7 is counterfeit.
- Every sack produces a different reading, from 551 to 560, so a single weighing always identifies the fake sack.
Why it fools people
Our instinct is to treat a weighing as a yes-or-no test, because that is how a balance scale works. A digital scale is far richer: it gives a number, and a number can encode a choice among many options. The solution works by giving each sack its own “signature” amount. Once you see weighing as encoding information, the answer feels almost obvious.
Did you get it?
About This Puzzle
An arcade operator in a beach town collects tokens from ten machines at the end of every summer week, one canvas sack per machine. This week, a tip comes in: someone has been feeding one machine with counterfeit tokens, and that whole sack is fake. The fakes look identical to the real ones. The only difference is weight. The operator has a precise digital scale, but the repair shop needs it back within the hour, so there is time for exactly one weighing. That sounds hopeless, since one weighing seems to test only one sack. A careful plan says otherwise.
How to Think About It
One reading on the scale is a single number, so the whole trick is to make that number carry as much information as possible. Think about what would happen if every sack contributed to the same weighing, but in a way that left a different fingerprint for each sack. If you put the same number of tokens from every sack on the scale, the fake sack would add the same extra weight no matter which one it was. So the contributions must differ. Ask yourself how to make the extra weight point directly to one sack.
Common Traps
Many people start by weighing one sack, or half of the sacks, as if this were a balance-scale puzzle. A balance only says heavier or lighter, but a digital scale gives an exact number, and wasting that precision is the first mistake. Others decide to weigh one token from each sack, which only confirms that a fake exists, something we already knew. The tempting assumption is that one weighing can test only one thing. It can test ten things at once if each sack is represented differently.
Where This Puzzle Comes From
Coin-weighing puzzles are among the oldest in recreational mathematics, and this digital-scale version has appeared in puzzle books and interview questions for decades. It is a gentle introduction to an idea that matters in computer science and testing: by combining many items in one measurement, with each item weighted differently, you can learn far more from a single test than you would by checking items one at a time.
Try This Variation
Now there are 5 sacks, and any number of them might be counterfeit, from none at all to all five. Fakes still weigh 11 grams and genuine tokens 10 grams. How can one weighing tell you exactly which sacks are fake?
Show the answer to the variation
Take 1, 2, 4, 8 and 16 tokens from sacks 1 to 5. The excess over 310 grams is a sum of distinct powers of two, and each total from 0 to 31 comes from exactly one set of sacks. An excess of 13 means 8 + 4 + 1: sacks 4, 3 and 1.