Perfect ruler

A perfect ruler of length ${\displaystyle n}$ is a ruler with a subset of the integer markings ${\displaystyle \{0,a_{2},\cdots ,a_{n}\}\subset \{0,1,2,\ldots ,n\}}$ that appear on a regular ruler. The defining criterion of this subset is that there exists an ${\displaystyle m}$ such that any positive integer ${\displaystyle k\leq m}$ can be expressed uniquely as a difference ${\displaystyle k=a_{i}-a_{j}}$ for some ${\displaystyle i,j}$. This is referred to as an ${\displaystyle m}$-perfect ruler.

A 4-perfect ruler of length ${\displaystyle 7}$ is given by ${\displaystyle \{0,1,3,7\}}$. To verify this, we need to show that every number ${\displaystyle 1,2,3,4}$ can be expressed as a difference of two numbers in the above set:

${\displaystyle 1=1-0}$

${\displaystyle 2=3-1}$

${\displaystyle 3=3-0}$

${\displaystyle 4=7-3}$

An optimal perfect ruler is one where for a fixed value of ${\displaystyle n}$ the value of ${\displaystyle a_{n}}$ is minimized.