Optimal Sparse Rulers

A normal ruler with lots of markings on it is actually overkill, you can make do with a smaller number of markings and still be able to measure any distance smaller than the length if the ruler itself.

-Alaric Stephen

I was inspired by Alaric Stephen's blogpost on minimal rulers to make a set of Optimal Sparse Rulers. These are the maximum lengths that 2 marks, 3 marks… 4, 5, 6, 7 marks can measure. 

A Perfect ruler has no repeating intervals. Unfortunately only the first 3 are perfect. (For example, on the 90mm ruler, you can measure 30mm between the 1cm mark and 4cm mark or the 4cm mark and 7cm mark).

As a side project, I tried creating an algorithm to generate these and found it fruitless. Maximal Sparse Rulers is an unsolved problem and greater mathematicians than I have tried and failed to create an algorithm for them.