Perfect ruler

A perfect ruler of length $\ell$ is a ruler with integer markings $a_{1}=0<a_{2}<\dots <a_{n}=\ell$ , for which there exists an integer $m$ such that any positive integer $k\leq m$ is uniquely expressed as the difference $k=a_{i}-a_{j}$ for some $i,j$ . This is referred to as an $m$ -perfect ruler.

An optimal perfect ruler is one of the smallest length for fixed values of $m$ and $n$ .

Example

A 4-perfect ruler of length $7$ is given by $(a_{1},a_{2},a_{3},a_{4})=(0,1,3,7)$ . To verify this, we need to show that every positive integer $k\leq 4$ is uniquely expressed as the difference of two markings:

1=1-0

2=3-1

3=3-0

4=7-3

Perfect ruler

Example

See also