MagicConstant 2006-11-17 17:47:09 2006-11-19 16:04:44 Definition magic square % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Given a magic square, magic cube, etc., the sum of any row, column or diagonal is called the \emph{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 $n^2$.