Ten Sacks, One Weighing
Ten sacks stand in a row, numbered 1 to 10. Nine of them are full of real coins weighing 10 grams each. One sack — you do not know which — is full of forgeries, and every forged coin weighs 9 grams: exactly one gram light.
This time the scale is a modern one with a display: it tells you a weight in grams, not just which side is heavier. Each sack holds hundreds of coins, and you may take out as many as you like from any sack.
Ten sacks. One is fake, and nothing on the outside says which.
Here is the beautiful part: it can be done, and the plan does not depend on luck at all. Work out what to put on the scale — then use the reading below.
Following your own plan, the display reads 543 grams.
Answer to type: the number of the fake sack.
💡 I want a nudge
If you take the same number of coins from every sack, the total tells you nothing about which sack it was. So take a different number from each.
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Answer: 7
- Take 1 coin from sack 1, 2 coins from sack 2, 3 from sack 3 … and 10 coins from sack 10. That is 1 + 2 + … + 10 = 55 coins in one heap.
- If every coin were real, the heap would weigh 55 × 10 = 550 grams.
- But the coins from the fake sack are 1 gram light each — and you took exactly as many of them as the sack's number. So the heap is short by exactly the number of the fake sack, in grams.
- The display reads 543. The shortfall is 550 − 543 = 7 grams.
- So the forgeries are in sack 7. One weighing, no luck involved.
The trick: a single number can carry a lot of information if you arrange for each possibility to change it by a different amount. Taking a different count from each sack turns "which one?" into arithmetic. This is exactly how a checksum works — one number that quietly tells you where something went wrong.