A perfect ruler of length ℓ {\displaystyle \ell } is a ruler with integer markings a 1 = 0 < a 2 < ⋯ < a n = ℓ {\displaystyle a_{1}=0<a_{2}<\dots <a_{n}=\ell } , for which there exists an integer m {\displaystyle m} such that any positive integer k ≤ m {\displaystyle k\leq m} is uniquely expressed as the difference k = a i − a j {\displaystyle k=a_{i}-a_{j}} for some i , j {\displaystyle i,j} . This is referred to as an m {\displaystyle m} -perfect ruler. An optimal perfect ruler is one of the smallest length for fixed values of m {\displaystyle m} and n {\displaystyle n} .
Example A 4-perfect ruler of length 7 {\displaystyle 7} is given by ( a 1 , a 2 , a 3 , a 4 ) = ( 0 , 1 , 3 , 7 ) {\displaystyle (a_{1},a_{2},a_{3},a_{4})=(0,1,3,7)} . To verify this, we need to show that every positive integer k ≤ 4 {\displaystyle k\leq 4} is uniquely expressed as the difference of two markings:
1 = 1 − 0 {\displaystyle 1=1-0}
2 = 3 − 1 {\displaystyle 2=3-1}
3 = 3 − 0 {\displaystyle 3=3-0}
4 = 7 − 3 {\displaystyle 4=7-3}
See also Golomb ruler Sparse ruler All-interval tetrachord This article incorporates material from perfect ruler on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.