Monthly Archives: April 2013

Gamma(1/4)

Posted on April 10, 2013 by apgoucher

The Gamma function, Γ, is an extension of the factorial function to the complex plane. Specifically, Γ(n) = (n−1)! for all positive integers n. It is defined, more generally, by analytic continuation of the following integral, which converges for all complex numbers … Continue reading →

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Cipher 24: ADFGVX

Posted on April 9, 2013 by apgoucher

For this week, I’ve decided to use a 95-year-old cipher employed by the German army during the First World War. You have the advantage, however, of the far greater computing power available today. FFXGDXXDG VVAFVXDDA DAVAXDFXG DVDAAAXFV AFAVGVVDV XGDXVXGAX AFDAVXADA … Continue reading →

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Don’t do it!

Posted on April 7, 2013 by apgoucher

Professor Sir Timothy Gowers FRS wrote about the usefulness of a particular proof strategy, known as ‘just do it’. He gives six example problems, providing proofs using this technique. I’ll now give alternative proofs, which use explicit constructions to avoid this idea completely. Problem … Continue reading →

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3D chess is Turing-complete

Posted on April 5, 2013 by apgoucher

As promised, here is the remainder of the proof of the Turing-completeness of three-dimensional chess. In the first part, we introduced the rules; in the second part, we built structures to function as logic gates and wires. Counter machines Instead of … Continue reading →

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BMO training

Posted on April 4, 2013 by apgoucher

Assuming everything has gone to plan, I am now introducing myself at the British Mathematical Olympiad training camp at Trinity. I’ll be assisting the future IMO team by making tea and providing biscuits (integral, if somewhat overlooked, roles in the administration … Continue reading →

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Circuitry in 3D chess

Posted on April 3, 2013 by apgoucher

This is the second of a projected three-part series of articles, which will ultimately prove the Turing-completeness of three-dimensional chess. In the first article, I described the basic rules of the game. In this article, I shall show how to … Continue reading →

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Cipher 23: Orthogonal boustrophedons

Posted on April 2, 2013 by apgoucher

This cipher is inefficient, in that it merely encodes 39 characters in the following (rather large!) matrix. Your challenge is to determine how this is done, and thus obtain the password.

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New prime-generating algorithm

Posted on April 1, 2013 by apgoucher

The usual method of generating the primes below N is to use a prime number sieve, such as the Sieve of Eratosthenes. This requires O(N log log N) operations for a random access machine, but can be reduced to O(N) … Continue reading →

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