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.

Perfect ruler

See also