## Puzzle #112: Weights and Balance Revisited

This is a twist to a puzzle which has been around for a very long time. I have myself looked at this puzzle since I was a kid, but a twist (or a potential flaw) to the puzzle was recently pointed out by Arushi (a 9th grader), a colleague’s daughter based in California. I was very impressed with Arushi’s thinking and am taking the liberty of sharing this with everyone. I am articulating the puzzle at three different levels (of varying difficulty):

1. Assume you have a digital scale. You want to be able to weigh (or see a reading of) whole numbers from 1 to 100 kg on the scale. You can have multiple different weights of any denomination you choose to, and can combine any subset of them to form each of the weight. What is the minimum number of weights you need (and in what denomination) to be able to measure all weights from 1 kg to 100 kg (only whole numbers)
2. Instead of a digital scale, now you have a balancing scale (i.e. you have two sides), and again you want to be able to measure all weights from 1 to 100 kg  – what is the minimum number of weights you need?
3. What if all you need is that if you have a package that weighs anywhere from 1 to 100 kg – only whole numbers, and your objective is that you should be able to tell what the weight is and you have a balancing scale – what is the minimum number of weights and what denominations do you need?

Please send your answers either directly on the blog site as comments, or to me at alokgoyal_2001@yahoo.com. If you like the puzzle, please share it with others. If you have interesting puzzles to share, please send them to me at my e-mail given above.

Happy balancing acts!

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