# magic constant

Given a magic square, magic cube, etc., the sum of any row, column or diagonal is called the *magic constant* of that magic square, cube, etc.

In the case of a standard $n\times n$ magic square that uses the integers from 1 to ${n}^{2}$, the magic constant is

$$\frac{1}{n}\sum _{i=1}^{{n}^{2}}i,$$ |

while that for a magic cube is

$$\frac{1}{{n}^{2}}\sum _{i=1}^{{n}^{3}}i.$$ |

We can then generalize to higher dimensions $d$ thus:

$$\frac{1}{{n}^{d-1}}\sum _{i=1}^{{n}^{d}}i.$$ |

So, for dimension $d$ the magic constant is $\frac{n({n}^{d}+1)}{2}$. For instance, a Franklin magic square ($n=8,d=2$) has magic constant $\frac{8({8}^{2}+1)}{2}=260$.

In a trivial sense, an $n\times n$ sudoku puzzle has a magic constant of ${n}^{2}$.

Title | magic constant |
---|---|

Canonical name | MagicConstant |

Date of creation | 2013-03-22 16:24:57 |

Last modified on | 2013-03-22 16:24:57 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 5 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 05B15 |