# perfect ruler

A perfect ruler  of length $n$ is a ruler with a subset of the integer markings $\{0,a_{2},\ldots,n\}\subset\{0,1,2,\ldots,n\}$ that appear on a regular ruler. The defining criterion of this subset is that there exists an $m$ such that any positive integer $k\leq m$ 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 $1,2,\ldots,4$ can be expressed as a difference of two numbers in the above set:

 $\displaystyle 1$ $\displaystyle=1-0$ $\displaystyle 2$ $\displaystyle=3-1$ $\displaystyle 3$ $\displaystyle=3-0$ $\displaystyle 4$ $\displaystyle=7-3$

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

Title perfect ruler PerfectRuler 2013-03-22 12:14:22 2013-03-22 12:14:22 mathcam (2727) mathcam (2727) 12 mathcam (2727) Definition msc 03E02 msc 05A17 Golomb ruler