Monthly Archives: March 2013

Three-dimensional chess

This is the first of a projected series of articles investigating a natural three-dimensional generalisation of chess. In the first article, I’ll briefly describe the rules and mention previous research in this area. The second article will show how to create a linear-bounded … Continue reading

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In 2010, Evangelos Kranakis and Danny Krizanc published an academic paper with a rather bizarre title. The Urinal Problem investigates a particular mathematical model arising from the behaviour of men selecting urinals in a bathroom arrangement. Rather atypical of mathematical publications, … Continue reading

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Cipher 22: There be dragons

This cipher has the interesting property that any vertical, horizontal or diagonal line hits an even number of characters. Of course, this piece of useless information won’t help you in the slightest.

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Rational distance problem

Suppose we have a unit square ABCD. Is it possible to place a point P in the plane of ABCD, such that PA, PB, PC and PD are all rational? It’s not too difficult to show that such a point … Continue reading

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Euler’s formula famously relates the number of vertices, edges and faces of a polyhedron. Specifically, it gives , where V, E and F are the numbers of vertices, edges and faces, respectively. For example, the dodecahedron has 20 vertices, 30 … Continue reading

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Cipher 21: Sudoku

You’ll need to solve a couple of NP-hard puzzles along the way. Enjoy! As usual, a password (which is entirely in lowercase) is included in the ciphertext, and allows you to access the secret area.

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Generalising Erdős’ conjecture

It’s a well-known fact that the harmonic series (i.e. the sum of the reciprocals of the natural numbers) diverges to infinity. There are at least two reasonably straightforward ways to prove this, the first being the Cauchy condensation test. Essentially, … Continue reading

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