Fewest weights when they may only go on one pan

Question

A grocer's balance has two pans. The item being weighed always sits alone on the left pan, and the grocer piles reference weights on the right pan until the two sides balance. The grocer wants to be able to weigh any item of 1, 2, 3, ..., up to 63 grams exactly.

What is the smallest number of reference weights that makes this possible, and which weights should they be?

(If you know the classic version where weights may go on either pan, notice how the answer changes when that freedom is taken away.)

Accepted Answer

Six weights: 1, 2, 4, 8, 16 and 32 grams. Why they work. Every whole number from 1 to 63 can be written in exactly one way as a sum of distinct powers of two: that is just its binary representation. To weigh 45 grams, for example, note 45 = 32 + 8 + 4 + 1 and put those four weights on the right pan. The largest total is 1 + 2 + 4 + 8 + 16 + 32 = 63, so the whole range is covered. Why fewer cannot work. With k weights, each weighing is a choice of which weights to use. Each weight is either on the pan or off it, giving at most 2 to the power k combinations, and one of those combinations is "no…

Fewest weights when they may only go on one pan

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