Perfect ruler

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A perfect ruler of length n is a ruler with a subset of the integer markings \{0,a_2,\cdots,a_n\}\subset\{0,1,2,\ldots,n\} that appear on a regular ruler. The defining criterion of this subset is that there exists an m such that any positive integer k\leq m can be expressed uniquely as a difference k=a_i-a_j for some i,j. This is referred to as an m-perfect ruler.

A 4-perfect ruler of length 7 is given by \{0,1,3,7\}. To verify this, we need to show that every number 1,2,3,4 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 n the value of a_n is minimized.

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This article incorporates material from perfect ruler on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.