There are 10 bags of coins. In 9 of them, each coin weighs 10 g. In the rest, all the coins weigh 11 g. You have only one weighing on a digital scale. How do you identify the different bag?
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Ten bags and a weighing
Ten bags and one weigh is one of those puzzles where the mindset matters more than the number of beads. The good solution does not add noise: it finds the idea that organizes the problem.
Hints
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- With a single weighing, the only option is to make each bag leave a different mark on the total.
- Don't take the same number of coins from each bag.
- If the final difference were 4 grams, that excess should point to a specific bag on its own.
Solution
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Answer: A single weighing is enough. Explanation: Take:
- 1 coin from bag 1,
- 2 from bag 2,
- 3 from bag 3,
- and so on up to 10 from bag 10. If they all weighed 10 g, the total would be
$$ 10(1+2+\cdots+10)=10\cdot 55=550\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\text{ g}. $$ But the defective bag has 11g coins. Therefore, the excess over 550 directly indicates what it is: - if the scale reads 553 g, the defective bag is number 3; - if it reads 557 g, it is number 7; - and so on. Each extra gram corresponds exactly to the number of coins taken from the weighed bag. $$
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