A perfect ruler of length

\ell

is a

ruler A ruler, sometimes called a rule, line gauge, or scale, is a device used in geometry and technical drawing, as well as the engineering and construction industries, to measure distances or draw straight lines. Variants Rulers have long ...

with

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

markings

a_1=0 < a_2 < \dots < a_n=\ell

, for which there exists an integer

m

such that any

positive integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

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

Example

See also