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