Monthly Archives: March 2013

Three-dimensional chess

Posted on March 30, 2013 by apgoucher

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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Urinals

Posted on March 28, 2013 by apgoucher

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

Posted on March 26, 2013 by apgoucher

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

Posted on March 24, 2013 by apgoucher

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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Holyhedra

Posted on March 21, 2013 by apgoucher

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

Posted on March 19, 2013 by apgoucher

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

Posted on March 18, 2013 by apgoucher

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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Three directions

Posted on March 15, 2013 by apgoucher

Firstly, I would like to announce that today is the Ides of March. Secondly, I was alerted on two semi-recent occasions to this beast of a puzzle, namely the regular expression crossword. Like a normal crossword, you populate the empty cells of a grid … Continue reading →

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Antoine’s necklace

Posted on March 15, 2013 by apgoucher

(Sorry about the recent dearth of cp4space postings; I’ve been rather busy in real Euclidean 3-space, .) Quite a large class of self-similar geometrical objects can be expressed as iterated function systems. For instance, the Sierpinski triangle is composed of three … Continue reading →

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Cipher 20: Ulam

Posted on March 12, 2013 by apgoucher

It’s difficult to believe that this is the twentieth cp4space cipher. Anyhow, here it is:  

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