Perfect ruler

A perfect ruler of length $n$ is a ruler with a subset of the integer markings $\{0,a_{2},\cdots ,a_{n}\}\subset \{0,1,2,\ldots ,n\}$ that appear on a regular ruler. The defining criterion of this subset is that there exists an $m$ such that any positive integer $k\leq m$ 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.

See also

• Golomb ruler
• Sparse ruler

This article incorporates material from perfect ruler on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.