The fewest weights puzzle
✓ With solution
A grocer wants to weigh every amount from 1 to 13 kg on a two-pan balance. How few weights can he manage with?
A grocer has a two-pan balance. He wants to buy a set of iron weights that lets him weigh any whole number of kilograms from 1 to 13 (1, 2, 3, … up to 13 kg). He can choose what each weight weighs, and he may put weights on both pans, even on the pan with the goods.
What is the smallest number of weights he needs?
- ATwo
- BThree
- CFour
- DFive
Show the answer
Answer: B) Three. Three are enough: 1, 3 and 9 kg. Each weight has three possible places: on the other pan, on the pan with the goods, or off the balance. For example, 2 = 3 − 1 (the 3 on one pan, the 1 with the goods), 7 = 9 + 1 − 3, 13 = 9 + 3 + 1, and so on for every amount from 1 to 13. Two weights can't do it: together they have only 3 × 3 = 9 arrangements, one of which weighs nothing, and the other eight come in mirror-image pairs, so two weights can weigh at most 4 different amounts. 1, 2, 4 and 8 would be four weights: that's the answer only if the weights must all go on one pan.
The grocer bought the three weights: 1, 3 and 9 kg. Then a customer asked for exactly 5 kg of sugar.
How does he weigh 5 kg of sugar with these weights?
Show a hint
You can't make 5 just by adding weights. Try putting some of them on the same pan as the sugar.
Show the answer
Answer: The 9 kg weight on one pan, the sugar with the 1 and 3 kg weights on the other. He puts the 9 kg weight on one pan and the 1 and 3 kg weights on the other, then pours sugar onto that second pan until the balance is level. The sugar is then 9 − 1 − 3 = 5 kg exactly.
Never miss a puzzle
Subscribe to Khamantiq on YouTube and turn on the bell to get new puzzles first.