The Market Scales Puzzle

At first I thought the four weights would be 1, 2, 4, 8. That’s the usual doubling pattern. But then I remembered this is a two-pan scale, so you can put weights on both sides. That means you can subtract as well as add. So the weights are 1, 3, 9, 27.

Why? Because with these four weights, you can make every number from 1 to 40 by putting some weights on one pan and some on the other. For example:

- 2 = 3 - 1

- 5 = 9 - 3 - 1

- 40 = 27 + 9 + 3 + 1

This works because every number from -40 to 40 can be written as a sum of ±1, ±3, ±9, ±27. It’s called balanced ternary, base 3, but the digits are -1, 0, and 1 instead of 0, 1, 2.

For a one-pan scale, you can only add weights, not subtract them. Then you need 1, 2, 4, 8, 16. These are powers of 2. With five weights, you can make every number from 1 to 31 by using each weight either once or not at all. That’s just binary representation. For example, 13 = 8 + 4 + 1.

This connects to ancient Egyptian multiplication. The Egyptians multiplied by doubling: 1, 2, 4, 8, 16, 32... That’s the same as the one-pan scale. So the one-pan puzzle is basically binary, and the two-pan puzzle is balanced ternary.

I think this would be a great activity for students. You could start with a smaller range, like weighing 1 to 4 grams with two weights (1 and 3). Let them figure out the pattern. Then ask what changes if you can only use one pan. It helps them see that number bases are not just abstract,  they come from real tools and real problems.

I also like that it connects to number theory. The weights are place values in a base system. Two-pan scale = balanced ternary. One-pan scale = binary. Same idea, different rules. It makes me want to try this with a class someday.

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