Perfect Ruler
A perfect ruler, also called a complete ruler, is type of ruler which has k distinct marks placed at integer distances from the end such that the distances between marks can be used to uniquely measure all the distances 1, 2, 3, 4, ... up to some maximum distance n>k. For example, the perfect difference set {1,2,5,7} gives 0, 1, 4, 6, which can be used to measure 1-0=1, 6-4=2, 4-1=3, 4-0=4, 6-1=5, 6-0=6, thus giving 6 distances using only four marks, as illustrated above (Gardner 1983, Fig. 91, and pp. 153-154).
No perfect Golomb ruler, i.e., a ruler that uniquely measures distances up to its length, exists for five or more marks (Golomb 1972; Gardner 1983, p. 154).
See also
Golomb Ruler, Perfect Difference Set, Ruler, Sparse RulerExplore with Wolfram|Alpha
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References
Gardner, M. Wheels, Life, and Other Mathematical Amusements. New York: W. H. Freeman, pp. 153-155, 1983.Golomb, S. W. "How to Number a Graph." In Graph Theory and Computing (Ed. R. C. Read). New York: Academic Press, pp. 23-37, 1972.Referenced on Wolfram|Alpha
Perfect RulerCite this as:
Weisstein, Eric W. "Perfect Ruler." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PerfectRuler.html