Author Archives: apgoucher

Some proofs of existence provide explicit examples. For instance, a proof of a composite Fermat number is as simple as noting that . On the other hand, some existence proofs are highly non-constructive, and do not provide explicit examples. The … Continue reading →

In an earlier cp4space post, I presented a set of five 5-sided dice. We can draw a directed graph associated with this set of dice: Each vertex represents a die. If die A beats die B with a probability greater … Continue reading →

In 1770, Leonhard Euler sent this particular curiosity to Joseph Lagrange. It’s a 4-by-4 magic square, all of whose entries are perfect squares. Martin Gardner offered a prize for finding a 3-by-3 magic square of squares. Lee Sallows found a … Continue reading →

A group* of twelve of us congregated in London to mark† the second round of the British Mathematical Olympiad. After several hours of marking, punctuated with a surprisingly high-quality lunch, we were able to announce the scores. The high scores are available on the BMOC website; well done … Continue reading →

Langton’s ant is a two-dimensional generalisation of a Turing machine, known as a turmite (a portmanteau of ‘Turing’ and ‘mite’, as well as being a pun on ‘termite’), renowned for its simple rules and interesting behaviour. Specifically, it has one state … Continue reading →

A positive integer N is described as a perfect number if the sum of all of its proper divisors (that is to say, factors of N other than N) is equal to itself. For instance, the proper divisors of 28 … Continue reading →

This is the 100th article on cp4space, and is inspired by a discussion last night with Gabriel Gendler. Triangle centres with complex numbers Suppose we have a triangle in the Euclidean plane . We can identify the Euclidean plane with the … Continue reading →

I’m going to start by describing a game that seems completely unrelated to sphere packing. The Conway-Hamming game involves a half-infinite row of green apples, each of which can either point up or down: In any configuration, all but finitely … Continue reading →