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I have looked at the interesting Conway-Guy Sequence which possesses a neat property of having unique subset sums. But I would like to find an integer set (which is optimally compact, ie has the smallest difference between the smallest and largest element) which possesses unique differences between every pair of numbers. I have wrote some python code that has brute forced the results up to set of size 9 but the code is exceedingly slow and I would like to know the optimal sets of about up to n=100 which is probably computationally impossible and requires an actual analytical solution. These are the results:

n=3 0,1,3
n=4 0,1,4,6
n=5 0,1,4,9,11
n=6 0,1,4,10,12,17
n=7 0,1,4,10,18,23,25
n=8 0,1,4,9,15,22,32,34
n=9 0,1,4,13,24,30,38,40,45
...

Also there are multiple sets that exist for each value of n which are optimally compact.

Mike Earnest
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Parsa IQT
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1 Answers1

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This is known as Golomb ruler. You can read about it in the Golomb ruler Wikipedia page. The Wikipedia page states that we don't know a lot about optimal Golomb rulers; the optimal solution for $n = 28$ was just confirmed in November 2022 after 8.5 years of calculation work.

VTand
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