Monthly Archives: December 2012

Recapping 2012

Posted on December 31, 2012 by apgoucher

Now that the year is drawing to a close, there are a few things worth discussing. Firstly, cp4space has a total of over 20000 views and a new seasonal banner (see above)! Mathematica 9 The Treefoil has been mentioned on mathpuzzle.com. … Continue reading →

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Busy beavers

Posted on December 30, 2012 by apgoucher

This is the third out of a series of four articles on increasingly fast-growing functions. The first article described the Ackermann function (corresponding to ω) and the Goodstein function (corresponding to ε_0). The second article went into much more detail about a … Continue reading →

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The inadequacy of SCLT

Posted on December 27, 2012 by apgoucher

As I mentioned a few posts ago, I included the Diophantine equation x^4 + y^6 = z^10 on the Advanced Mentoring Scheme. I’m not going to spoil it here, although I have since been informed that I had previously included it … Continue reading →

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Cipher 9: Christmas cryptography

Posted on December 25, 2012 by apgoucher

Being simultaneously Christmas Day and Cipher Tuesday, I have a lot of material to get through. Isaacs Firstly, happy 370th birthday to Sir Isaac Newton, who succeeded Isaac Barrow as the Lucasian Professor of Mathematics (both from Trinity, yay!). This gives … Continue reading →

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The world still exists

Posted on December 22, 2012 by apgoucher

It transpires that the world didn’t actually end yesterday. At the very least, Descartes’ famous deduction ‘cogito ergo sum’ seems to imply that. To summarise, the Mayan calendar has finished its 13th long count cycle; equivalently, 13×20×20×18×20 days have passed since … Continue reading →

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Dissecting the disc

Posted on December 20, 2012 by apgoucher

At the tenth Gathering for Gardner, Colin Wright proposed the following problem. It’s quite well known, and I believe it has been published elsewhere before: ‘Dissect a [unit] disk into congruent parts at least one of which avoids the center by … Continue reading →

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TREE(3) and impartial games

Posted on December 19, 2012 by apgoucher

This article was originally supposed to be about TREE(3) and the busy beaver function. However, I realised the potential of turning TREE(3) into a two-player finite game, which is surprisingly fun and means that I’ve ended up leaving uncomputable functions until a later post. … Continue reading →

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Cipher 8: Honeycomb

Posted on December 18, 2012 by apgoucher

The inspiration for this cipher stemmed from a conversation with James Aaronson, when we considered the prospect of a whole new category of cipher. I was initially sceptical as to whether it could actually be implemented (they’re certainly much harder … Continue reading →

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Pictures of matchstick graphs

Posted on December 17, 2012 by apgoucher

A matchstick graph is a planar graph with a plane embedding where all edges are of unit length. The name derives from the fact that they can be assembled on a flat surface out of matchsticks of equal length. Of … Continue reading →

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Fast-growing functions

Posted on December 15, 2012 by apgoucher

This is the first of a projected two-part series of articles about fast-growing functions. The first part (‘fast-growing functions’) will introduce the concept of a fast-growing hierarchy of functions, use some notation for representing large numbers, make an IMO shortlist problem infinitely more … Continue reading →

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