Problem 1428. Find the optimal shape to bring the maximum product by a given perimeter

Created by Eliad
Appears in Computational Geometry II

Difficulty:Rate

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Find the optimal shape (with Nopt sides, where Nopt is at least 3) to bring the maximum product of the sides length, given a specified perimeter.

Example: Assuming perimeter equals to 7 (p=7) then the known optimal value is Nopt=3 (a triangle, so 3 sides). let mark L1,L2,L3 the triangle sides length (L1+L2+L3=7), thus the product is L1*L2*L3 and we want the product to be as large as possible for your chosen number of sides Nopt.

Don't assume the perimeter is an integer. The side lengths need not be integers, although it is possible they might be so. And of course, you need to decide if the optimum happens for an equilateral polygon, so with equal side lengths, or if some other set of side lengths that collectively sum to p might be a better choice.

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Difficulty level on a scale 1 to 10: 8

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159 Solutions
42 Solvers

Last Solution submitted on Sep 09, 2023

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John D'Errico on 26 Mar 2018

I took a look at the test cases. The 4th test has a result that is expected to be on the order of 1e40. Then it compares that to the solution returned, and tests to see if the absolute difference is less then 1e-4? eps(1e40) is roughly 1e24. So the difference between two numbers of size 1e40 will never be less than 1e-4, unless they are IDENTICAL. Far better would be to test if the relative difference is small.

John D'Errico on 26 Mar 2018

I updated the problem explanation to be a bit more clear. I also repaired the checks for a valid solution to use a relative test, mainly valuable for one of the cases where an absolute test on the error was a poor choice.

J. S. Kowontan on 12 Apr 2018

Clearly, the best score comes from older solvers not having to deal with singularity that occurs when p = 5, which was not included in the test suite back then.

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