Four weights that measure everything up to 40

Question

A merchant has a two-pan balance and wants to be able to weigh out any whole number of pounds from 1 to 40. To do it, they may place their reference weights on either pan, including the same pan as the goods.

What four weights let the merchant measure every integer load from 1 to 40 pounds?

Accepted Answer

The four weights are 1, 3, 9, and 27 pounds (the powers of 3). Notice 1 + 3 + 9 + 27 = 40, so 40 is the most they can ever total. Why placing weights on both pans is the key. Because a weight can go on the goods' side, each weight can effectively be added to, subtracted from, or left out of the measurement, three roles. Put a weight on the opposite pan from the goods and it counts +; put it on the same pan as the goods and it counts -; leave it off and it counts 0. This is base-3 with digits -1, 0, +1 ("balanced ternary"). Every integer from 1 to 40 can be written as a sum of 1, 3, 9, 27 each…

Four weights that measure everything up to 40

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