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Balance Scale Puzzle

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Balance Puzzle - Polypad - Mathigon
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https://mathigon.org/polypad/9K5Wg0rE8hJdw

Instructions

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?

  1. You may use objects by assigning values as well as the numbers you have picked directly. 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.
  1. You need to balance all the numbers(weights) from 1-40 by using the set of numbers (weights) you choose.
  1. For an extra challenge, which weight would you add to measure the weights of objects heavier than 40kg.

Videos

This puzzle is also mentioned by Marcus du Sautoy on Numberphile:

Mathematics is all about SHORTCUTS - Numberphile
Featuring Marcus du Sautoy from Oxford University.Check his latest book "Thinking Better: The Art of the Shortcut"Links & stuff in full description below ↓↓↓...
https://www.youtube.com/watch?v=BdEWCxt8C0M&feature=emb_imp_woyt

Solution

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.