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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Famous quotes containing the words perfect and/or ruler:
“...I never drink wine ... I keep my hands soft and supple ... I sleep in a soft bed and never over-tire my body. It is because when my hour strikes I must be a perfect instrument. My eyes must be steady, my brain clear, my nerves calm, my aim true. I must be prepared to do my work, successfully if God wills. But if I perish, I perish.”
—Lisa, Russian terrorist (anonymous)
“Can anything be more ridiculous than that a man should have the right to kill me because he lives on the other side of the water, and because his ruler has a quarrel with mine, though I have none with him?”
—Blaise Pascal (1623–1662)