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Find four weights such that given four weights and weighing pan (balance scale) you can measure all weights between $1$ to $80$.

I found this one here.Any idea how to solve?

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    The problem requires some clarification. It's sometimes asked how 4 weights can be used to measure all weights between 1 to 40. The intended meaning there is that each of these must be achievable as the difference between the sums of weights on the two scales. With that meaning of "measure", it's impossible to do the same for 1 to 80. I think the question means: Find 4 weights such that if you know that what you're measuring is an integer between 1 and 80 you can always infer the given weight. Brute search reveals that in addition to the guessable answer there's a slightly different one, too.2011-02-08
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    That alternative answer is actually a consequence of the problem not being posed economically -- the guessable answer allows you to infer the weight if you know it's between 1 and *81*, whereas the modified answer only works up to 80.2011-02-08
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    ... is your "alternative answer" just "the regular answer with the heaviest weight reduced by 1"? If so, I spent far too long wondering about that...2011-02-08
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    @Rawling: It is, yes -- sorry about that! I did say it was "slightly different" :-)2011-02-08
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    Use the integer closest to $e=2.718...$ as a base, and hence as suggested use $3$.2011-02-08
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    @Eric Naslund: Not sure whether that was a humorous remark? :-) That's the optimal base for FFT -- are you saying a similar argument applies here? I think 3 is the exact optimal base for this case, determined by the fact that there are 3 different things we can do with a weight, corresponding to changing the balance by +1, 0 or -1 times its weight.2011-02-08

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