Maths prize for 3 November 2015: weighing

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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?

The prize was won by Khaleah Edwards.


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 3k weight and you’re able to measure weights up to ½(3k+1−1)

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