You have a scale with two pans and four weights:
$$ 1,\ 3,\ 9,\ 27\ \text{kg}. $$
You can put weights on either side.
How many different whole weights between 1 and 40 kg can you measure exactly?
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The scales of Babel
Hints
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- With weights on both plates, each weight can contribute: +w (on the weight side).
- With $w\in\{1,3,9,27\}$ this is equivalent to representing the searched weight in balanced base 3 with digits $-1,0,+1$.
- All integers from 1 to 40 (40 values) can be measured.
Solution
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Answer: all integers from 1 to 40 (40 values) can be measured.
With weights on both plates, each weight can provide:
- $+w$ (on the weight side),
- $0$ (do not use),
- $-w$ (on the object side).
With $w\in\{1,3,9,27\}$ this is equivalent to representing the desired weight in balanced base 3 with digits $-1,0,+1$.
The absolute maximum representable is:
$$ 1+3+9+27=40. $$
And balanced base 3 guarantees coverage of all integers between $-40$ and $40$. In particular, all positive weights from 1 to 40.
Example:
$$ 20 = 27-9+3-1. $$
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