A perfect ruler of length is a ruler with integer markings , for which there exists an integer such that any positive integer is uniquely expressed as the difference for some . 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 . To verify this, we need to show that every positive integer is uniquely expressed as the difference of two markings: See also Golomb ruler Sparse ruler All-interval tetrachord Combinatorics