perfect ruler

A perfect ruler of length $n$ is a ruler with a subset of the integer markings $\{0,a_2,\mathrm{\dots },n\}\subset \{0,1,2,\mathrm{\dots },n\}$ that appear on a regular ruler. The defining criterion of this subset is that there exists an $m$ such that any positive integer $k\le m$ can be expresses uniquely as a difference $k=a_i-a_j$ for some $i,j$. This is referred to as an $m$-perfect ruler.

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

$1$$=1-0$

$2$$=3-1$

$3$$=3-0$

$4$$=7-3$

An optimal perfect ruler is one where for a fixed value of $n$ the value of $a_n$ is minimized.