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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 4Space. 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 fourdimensional 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 600cell 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 bullhuman 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 lifesize bronze head of a philosopher. The most valuable treasure, however, was a 2000yearold 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 nontrivial 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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Linearbounded automata
If we provide a bogstandard 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 email 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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