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.
A perfect ruler that can measure up to its own length is called a sparse ruler. A few perfect rulers can measure longer distances than an optimal sparse ruler with the same number of marks., and can each measure up to 18, while an optimal sparse ruler with 7 marks can measure only up to 17. Likewise, can measure up to 24, which is better than the 23 for an optimal sparse ruler with 8 marks.
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