Monthly Archives: September 2012

Hiatus interruptus

I shall be rather busy during the next couple of months, so there may be a noticeable drop in the frequency of posts on Complex Projective 4-Space. Nevertheless, I shall endeavour to maintain a steady stream of interesting articles, assuming I … Continue reading

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Good fibrations

We’ll begin by considering Hamiltonian quaternions, a four-dimensional extension to the complex numbers. A particularly interesting bunch of quaternions is the multiplicative group of 120 ‘icosians’. These can be thought of as the vertices of a 600-cell or, by duality, the … Continue reading

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Cipher 2: Labyrinth

The Labyrinth usually refers to a particular maze in Knossos on the Greek island of Crete, designed by Daedalus to contain the Minotaur: a ravenous bull-human hybrid, which found human sacrifices particularly appetising. Theseus navigated these tortuous underground passageways by means of … Continue reading

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MODA: Inequalities

I imagine you’re all eagerly anticipating Cipher Tuesday, which is only a couple of days away now! Meanwhile, the sixth chapter of MODA, namely ‘Inequalities’, is available for download. Preface Combinatorics I (enumerative and geometrical combinatorics) Linear algebra Combinatorics II (graph theory) Polynomials Sequences Inequalities Projective geometry … Continue reading

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163

There is a reasonably popular card game entitled 24, where players attempt to derive 24 by applying basic arithmetic operations to a set of four positive integers. For example, the provided integers may be {3, 3, 8, 8}. A solution must … Continue reading

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Antikythera mechanism

Near the beginning of the 20th century, a shipwreck was discovered. It contained a horde of treasure en route from Greece to Rome, including a life-size bronze head of a philosopher. The most valuable treasure, however, was a 2000-year-old analogue … Continue reading

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MODA: Sequences and Polynomials

It’s reached this time of week again, so I’ve uploaded the fourth and fifth chapters of Mathematical Olympiad Dark Arts. Enjoy! Preface Combinatorics I (enumerative and geometrical combinatorics) Linear algebra Combinatorics II (graph theory) Polynomials Sequences Inequalities Projective geometry Complex numbers Triangle … Continue reading

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Lizard and Spock dice

Continuing the theme of magic squares, we begin with the unique simplest non-trivial magic square. It contains the integers {1, 2, …, 9}, one in each square, such that the rows, columns and diagonals all sum to 15. Remarkably, the converse is … Continue reading

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Linear-bounded automata

If we provide a bog-standard computer with an infinite data store, it suddenly becomes a Turing machine: capable of answering any question answerable by any digital computer. Even quantum computers are no more powerful; they are merely faster. For example, a quantum … Continue reading

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Lunisolar calendars

Here’s the post I promised you a while ago. I wrote the first part of this in an e-mail on Burns’ Night of 2011, and the main idea (which Alex Bellos termed the ‘Goucherian calendar’) was mentioned in the Guardian. … Continue reading

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