Tag Archives: recreational maths

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.

Numbers and Logic

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I am a big fan of number theory. I find the answer to Hilbert’s Tenth Problem fascinating. I was introduced to this problem, a couple of years ago, via the documentary titled : “Julia Robinson and Hilbert’s Tenth Problem“, here is the trailer:

 

You can read more about it here. Also for the sake of completeness, let me state Hilbert’s Tenth Problem:

Does there exist an algorithm to determine whether a given Diophantine equation has a solution in rational integers?

In 1970, Yuri Matiyasevich completed the solution of this problem by using the concept of Turing Machine. This short video provides a nice overview about Turing Machines in general

The answer to Hilbert’s Tenth Problem problem is

No such algorithm exists.

This interplay of number theory and logic is really interesting, isn’t it? But I can’t discuss solution of Hilbert’s Tenth Problem here, since I have never read it. But there is nice overview at Wikipedia.

I will rather discuss a puzzle from Boris A. Kordemsky’s book which illustrates the idea of this interplay.

Ask a friend to pick a number from 1 through 1000. After asking him/her ten questions that can be answered yes or no, you tell him/her the number. What kind of question?

The key to the solution is that 2 to the tenth power is 1024 (that is, over 1000). With each question you knock out half the remaining numbers, and after ten questions only the thought number is left.

I welcome you to think of a number and write the corresponding yes/no questions as a comment below.