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 languag ...

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 Difference, The Difference, Differences or Differently may refer to: Music * ''Difference'' (album), by Dreamtale, 2005 * ''Differently'' (album), by Cassie Davis, 2009 ** "Differently" (song), by Cassie Davis, 2009 * ''The Difference'' (al ...

k=a_i-a_j

for some

i,j

. This is referred to as an

m

-perfect ruler. An

optimal Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

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