# perfect ruler

A *perfect ruler ^{}* of length $n$ is a ruler with a subset of the integer markings $\{0,{a}_{2},\mathrm{\dots},n\}\subset \{0,1,2,\mathrm{\dots},n\}$ that appear on a regular ruler. The defining criterion of this subset is that there exists an $m$ such that any positive integer $k\le m$ 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 $1,2,\mathrm{\dots},4$ can be expressed as a difference of two numbers in the above set:

$1$ | $=1-0$ | ||

$2$ | $=3-1$ | ||

$3$ | $=3-0$ | ||

$4$ | $=7-3$ |

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 |