perfect ruler

A perfect ruler of length $n$ is a ruler with a subset of the integer markings $\{0,a_{2},\ldots,n\}\subset\{0,1,2,\ldots,n\}$ 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 expresses 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 $1,2,\ldots,4$ can be expressed as a difference of two numbers in the above set:

	$\displaystyle 1$	$\displaystyle=1-0$
	$\displaystyle 2$	$\displaystyle=3-1$
	$\displaystyle 3$	$\displaystyle=3-0$
	$\displaystyle 4$	$\displaystyle=7-3$

An optimal perfect ruler is one where for a fixed value of $n$ the value of $a_{n}$ is minimized.

Title	perfect ruler
Canonical name	PerfectRuler
Date of creation	2013-03-22 12:14:22
Last modified on	2013-03-22 12:14:22
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	12
Author	mathcam (2727)
Entry type	Definition
Classification	msc 03E02
Classification	msc 05A17
Synonym	Golomb ruler