Ten bags and one weigh is one of those puzzles where the way you ask the question matters more than the amount of calculations. The elegant solution does not multiply operations: it finds the idea that organizes the problem in a single action.
Ten bags and one heavy
Riddle statement
There are 10 bags of coins. In 9 of them, each coin weighs 10 g. In the remaining one, all the coins weigh 11 g.
You have only one weighing on a digital scale.
How do you identify the different bag?
Show solution
Solution
Answer: A single weighing is enough.
Procedure:
Take:
- 1 coin from bag 1,
- 2 from bag 2,
- 3 from bag 3,
- and so on up to 10 from bag 10.
If all the coins weighed 10 g, the total would be
$
10(1+2+\cdots+10)=10\\times 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\text{ g}.
$
The defective bag contains 11 g coins, so the excess over 550 g directly indicates which one it is:
- if the scale reads 553 g, the heavy 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 bag with the most weight.