A perfect ruler of length is a ruler with a subset of the integer markings that appear on a regular ruler. The defining criterion of this subset is that there exists an such that any positive integer can be expressed uniquely as a difference for some . This is referred to as an -perfect ruler.
A 4-perfect ruler of length is given by . To verify this, we need to show that every number can be expressed as a difference of two numbers in the above set:
An optimal perfect ruler is one where for a fixed value of the value of is minimized.