You have a scale with two pans, so for example you can measure 2 grams exactly by putting a one-gram weight in one pan and a three-gram weight in the other. With what four weights can you measure any weight up to 40 grams? With what five can you measure any weight up to 121 grams?

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Solution: 1 and 3 weigh anything up to 4 grams.

Once you have weights to weigh anything up to n grams, add a 2n+1 weight.

For n+1 grams, you put 2n+1 in one scale, and add whatever makes n on the opposite side; for n+2, 2n+1 in one scale, and add whatever makes n−1, and so on.

For 2n+2, you put 2n+1 in one scale, and 1 in the same scale. For 2n+3, you put 2n+1 in one scale, and add whatever makes 2 on the same side (i.e. 3 in the same scale, 1 in the opposite one). And so on up to 3n+1.

So 1 and 3 give n=4

Add 9 (2×4+1) and you get n=13

Add 27 (2×13+1) and you get n=40

Add 81 (2×40+1) and you get n=121

At each stage you add a 3^{k} weight and you’re able to measure weights up to ½(3^{k+1}−1)

This is called Bachet’s problem, and it is four hundred years old.

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