Monthly Archives: August 2013

Four colour theorem

A fascinating paper by Louis Kauffman establishes the equivalence of the four-colour theorem (the assertion that any planar graph can have its vertices coloured with one of four colours, such that neighbouring vertices have different colours) to the following seemingly unrelated combinatorial … Continue reading

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Comfortably squarefree numbers

Usually, it is very difficult to find a reasonably naturally-defined sequence which is not present on the Online Encyclopedia of Integer Sequences. The OEIS contains everything from the ubiquitous Bernoulli numbers and the fascinating Golomb self-referential sequence, to such utterly pointless trivialities as A171056 (the … Continue reading

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Stella octangula

In this rather popular video, Vi Hart experiments with the patterns that can be formed by connecting the vertices of a regular polygon with straight lines of equal length. In addition to the obvious number-theoretic interpretation, there is a nice group-theoretic … Continue reading

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Cipher 42: Sudoku II

This can be regarded as a sequel to cipher 21. One of the problems with that cipher was the fact that people could cheat by using an online Sudoku solver; I don’t believe that this suffers from the same vulnerability. … Continue reading

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A more aperiodic monotile

This article can be regarded as a sequel to my 5 × 5 monotile. Since its announcement, several variants have been discovered. In particular, towards the end of this article we shall present a three-dimensional monotile with the following properties: Nonperiodic by … Continue reading

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Block cellular automata

On an infinite square lattice, the B36/S125 cellular automaton proceeds similarly to Conway’s Game of Life (B3/S23), but with different birth and survival conditions. Specifically, a dead cell becomes live if surrounded by 3 or 6 live neighbours, and a … Continue reading

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New aperiodic monotile

In three dimensions, an aperiodic monotile is a solid capable of tiling space, but not in such a way that admits translational symmetry. The question of existence of aperiodic monotiles stems from a weaker question, which formed part of Hilbert’s … Continue reading

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