# Magic Cubes

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Last week I attended a talk (by a student) about Magic Squares. I learned a bunch of cool facts about them (like how to devise an algorithm to construct them). Towards the end of the talk, one student from the audience suggested the possibility of Magic Cubes. I got very excited about this idea since it pointed towards the stereotypical mathematical ideology of generalizing the examples in order to see the deeper connections.

I myself don’t know much about Magic Cubes (or even Magic Squares) but would like to quote W. W. Rouse Ball & H. S. M. Coxeter from pp. 217 the book “Mathematical Recreations and Essays” (11th Ed.) :

A Magic Cube of the $n^{th}$ order consists of the consecutive numbers from 1 to $n^3$, arranged in the form of a cube, so that the sum of the numbers in every row, every column, every file, and in each of the four diagonals (or “diameters “), is the same-namely, $\frac{1}{2}(n^3 + 1)$. This sum occurs in $3n^2 + 4$ ways. I do not know of any rule for constructing magic cubes of singly-even order. But such cubes of any odd or doubly-even order can be constructed by a natural extension of the methods already used for squares.

I would like to read about these magic hyper-cubes in future. And if you know something interesting about them, let me know in the comments below.

Anyways, I found it pretty amazing to discover that there is an existence condition for k>1 and $n \neq 2$. And the proof of it apparently provides a way to always construct a magic hypercube of the given dimension and order. But the biggest question of all would be to list the set of all possible magic hypercubes, given order and dimension. I don’t know if some work has already been done in this light.