Perfect ruler

A perfect ruler of length $n$ is a ruler with a subset of the integer markings $\{0,a\_2,cdots,n\}subset\{0,1,2,ldots,n\}$ that appear on a regular ruler. The defining criterion of this subset is that there exists an $m$ such that any positive integer $kleq\; m$ can be expressed 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 $1,2,3,4$ 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.

ee also

*Golomb ruler