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:
“It is best for all parties in the combined family to take matters slowly, to use the crock pot instead of the pressure cooker, and not to aim for a perfect blend but rather to recognize the pleasures to be enjoyed in retaining some of the distinct flavors of the separate ingredients.”
—Claire Berman (20th century)
“For in the division of the nations of the whole earth he set a ruler over every people; but Israel is the Lord’s portion: whom, being his firstborn, he nourisheth with discipline, and giving him the light of his love doth not forsake him. Therefore all their works are as the sun before him, and his eyes are continually upon their ways.”
—Apocrypha. Ecclesiasticus 17:17-9.