A perfect ruler of length

\ell

is a

ruler A ruler, sometimes called a rule, scale, line gauge, or metre/meter stick, is an instrument used to make length measurements, whereby a length is read from a series of markings called "rules" along an edge of the device. Usually, the instr ...

with

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

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 the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...

k\leq m

is uniquely expressed as the

difference Difference commonly refers to: * Difference (philosophy), the set of properties by which items are distinguished * Difference (mathematics), the result of a subtraction Difference, The Difference, Differences or Differently may also refer to: Mu ...

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 criteria, from some set of available alternatives. It is generally divided into two subfiel ...

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