Perfect ruler explained

A perfect ruler of length

\ell

is a ruler with integer markings

a₁₌₀<a₂<...<a_n=\ell

, for which there exists an integer

such that any positive integer

k\leqm

is uniquely expressed as the difference

k=a_i-a_j

for some

i,j

. This is referred to as an

-perfect ruler.

An optimal perfect ruler is one of the smallest length for fixed values of

and

Example

A 4-perfect ruler of length

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\leq4

is uniquely expressed as the difference of two markings:

1=1-0

2=3-1

3=3-0

4=7-3