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Create a set of weights so that any object with an integer weight from 1 to 40 kilograms can be balanced on a two-sided scale. What is the fewest number of weights you need, and what are the weights?
For example, if you choose 1, 3, 5, and 10 as the set of weights, list them under the weight object to start trying all the numbers.
You need to balance all the numbers(weights) from 1-40 by using the set of numbers (weights) you choose. For an extra challenge, find out which weight would you add to measure the weights of objects heavier than 40kg.
This puzzle is also mentioned by Marcus du Sautoy on Numberphile:
For younger students;
Invite them to prove that all weights 1-40 kg can be balanced using only 1, 3, 9, and 27 kilograms. Use notice and wonder routine to discuss their findings. If does not come up naturally, ask, which weight would they add to measure the weights heavier than 40kg.
The set of weights you need to choose is 1, 3, 9, and 27. For the challenge to weigh objects heavier than 40 kg, you need to add 81 kg.