For some reason, my PhD thesis has somehow never become freely available online – so here it is.

My favourite part is the presentation of Theorem 4.7.3 as a two-player game – this and the wiggly diagrams is almost exactly how I reasoned through this problem (plus a key insight of the proof was written on a serviette during a pub quiz).

I would always tell my undergraduate students that my favourite series was the harmonic series to help them remember it, and because it acts as a counter-example to so many close-to-seeming true conjectures. The proof of the above theorem makes key use of a shifted version of the alternating harmonic sequence.

If I were to continue researching, obviously Chapter 7 “Further work” was a starting point (and indeed my last unpublished paper built on top of that). Looking at how methods of generalising norm constructions on top of other norms (mapping norms to other norms) give rise to various Banach sequence space properties. I looked into ‘Jamesifying’ other spaces.

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Arithm’ and clues in Puzzlebomb

I had my second puzzle in the January 2015 online puzzle sheet Puzzlebomb, entitled “Arithm’ and clues“. It’s a series of cryptarithm puzzles with the same letters denoting the same numbers throughout, but with none of the words given: instead they’re clued. It’s like a crossword but with arithmetic replacing the grid.

Go on and give it a try before reading on, as there are some tiny hints ahead. Continue reading

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About time

Sometimes I start newcomers to MathsJam on timing puzzles. Here are five classics in increasing order of difficulty and implausibility, each with a distinct flavour. The first is distinctly steaky…

1. Some friends are coming over for a steak dinner. You want to cook three steaks as quickly as possible, but your grill pan only holds two at a time. Each steak must be cooked for five minutes on each side. What is the fastest you can have all three ready? Show that you can’t do any better.

Alas, when this precise situation once arose with my family at home (who says puzzles are never useful?), I was unable to convince my father not to cook the steaks in the obvious order. But at least that meant I got my steak sooner.

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More on Gale–Shapley Nobel prize

I’ve long been a fan of the Gale–Shapley matching algorithm, and related problems, so was happy to see that a Nobel Prize was awarded for it. Having seen Peter Rowlett’s article that laid down the following gauntlet:

“I see ‘Nobel week’ as an opportunity for mathematicians to go in search of the mathematics behind each prize, rather than to retreat and complain about the lack of a prize specifically for mathematics”,

I was surprised that none of the mathsy types in my tiny corner of internet seemed to have noticed that a mathematician won a Nobel prize essentially for mathematics. After growing slightly impatient, I realised I only had myself to blame for not acting earlier, so I sketched a quick news story contribution over at the Aperiodical (it’s short and so reproduced here in full):
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Solution to In Their Prime

Did you have a go at my In their Prime puzzle in Puzzlebomb yet? How about the other puzzles?

If your answer to these questions is “No”, then please turn to  page 13091204281 of the internet to have a go at the July Puzzlebomb.

If the answer is “Yes”, you can check the July solutions, including the numeric solution to In their Prime. But how did you find the solution? You may have successfully used trial and error, but that’s not usually very enlightening; you may have programmed a computer to do the dirty work for you (the puzzle was hand designed, but I did check the answer was unique with a bit of code). As a responsible puzzle-setter, I came up with the following possible proof of the solution. I’d be interested to hear from anyone who had a different method.

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In Their Prime in Puzzlebomb

I’ve contributed a one-off family tree puzzle called “In Their Prime” to the July issue of the excellent monthly puzzle publication Puzzlebomb. It’s about an extended family who die when they reach a semiprime (composite of two primes) age determined by their parents’ ‘prime number’ genes. My aim was to design a puzzle with a “How do I get started?” flavour which probably would be lost if there was a sequel without a sufficiently interesting twist.

If you haven’t already read them, there are two posts on the Aperiodical by Paul Taylor explaining the computer creation of and maths behind two puzzles that have also featured in Puzzlebomb:

  • The extremely unique fractalphile “Hilbert’s Space-Filling Crossword” (only one non-trivial such puzzle exists).
  • The more abundant “Spelling Bees” which have appeared several times in past issues (May, June and the aforementioned July issue), involve finding Hamiltonian paths that spell out a pair of words or phrases.

Also in the May issue, I especially liked “Word Split” a pentomino-based colouring word search (but I cheated by not breaking out the crayons). All issues can be found on the Puzzlebomb section of the Aperiodical.

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Post on Aperiodical

I’ve written an article for the Aperiodical entitled “Ask a mathematician: where should we live?”.

Dear Mathematician,

My partner and I are trying to buy a house. We both work in different places, and neither of us enjoys commuting. How could we decide where to live?

Fictionally yours,

Norman Mettrick


Thank you for your intriguing and entirely imaginary letter. The short and not terribly useful answer would be…

Want to know? Read the rest of it there.

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Jozef Schreier, Schreier sets and the Fibonacci sequence

This post has some more technical bits, as it’s aimed mainly at people with an undergraduate-level background in mathematics. I’ve included a historical note on Schreier, because it’s easy to forget that mathematical objects are named after real people.

These five sets (each of which have their elements written in numerical order) are Schreier sets:


The following six sets are not Schreier sets:



Can you spot the simple rule that defines whether a set of numbers is a Schreier set or not?

Historical Background

Jozef Schreier was born 1908/1909 in Lwów (then part of the Austro-Hungarian empire, now in Ukraine) and later based at the university there (whilst it was part of Poland). Schreier was killed in April 1943 during the Second World War in his home town of Drohobycz. In his book “Adventures of a Mathematician“, Stanislaw Ulam wrote about Schreier, his friend and collaborator:

We would meet almost every day, occasionally at the coffee house but more often at my house. His home was in Drohobycz, a little town and petroleum center south of Lwów. What a variety of problems and methods we discussed together! Our work, while still inspired by the methods then current in Lwów, branched into new fields: groups of topological transformations, groups of permutations, pure set theory, general algebra. I believe that some of our papers were among the first to show applications to a wider class of mathematical objects of modern set theoretical methods combined with a more algebraic point of view. We started work on the theory of groupoids, as we called them, or semi-groups, as they are called now.

Schreier was also a major contributor (ten questions) to the now famous Scottish book (an important book of questions, famous at least amongst Banach space theorists, and which you can read in English); in fact, he and Ulam pair were the only undergraduate student participants in the Scottish Cafe, a coffee shop where mathematicians from Lwów met, and scribbled maths directly onto the marble tabletops. The Scottish book was a notebook to save the work from being cleaned away at the end of the day.

The Baire-Schreier-Ulam and Schreier-Ulam theorems both bear Jozef Schreier’s name. He has on occasion been referred to as “Julian Schreier” (possibly by conflating him with Juliusz Schauder?): whereas Ulam’s account, the Portugese Wikipedia article, which seems to have been written by his family, and this journal from the time all name him Jozef (or a variant: Josef/Józef/Joseph). He definitely shouldn’t be confused with the contemporary mathematician Otto Schreier, who specialised in groups.

One of Jozef Schreier’s conjectures is now referenced in computer science literature: inspired by an article of Lewis Carroll, in the 1932 paper “On tournament elimination systems“, he conjectured that to find the second largest number in an unordered list requires at least n + \lceil \log_2 n \rceil -2 comparisons, which was later confirmed in the 1960s.

I’ve included a list of some of Schreier’s publications at the end of this article.

Schreier sets

In 1930, Schreier constructed a counterexample to a question of Banach and Saks using the following idea:

A (non-empty) subset X of the natural numbers is a Schreier set if its size is not larger than its least element.

|X| \leq \min X

The collection of all such subsets of \mathbb{N} is known as the (first) Schreier family. For example: \{1\}, \{2\} and \{3,5,42\} are all Schreier sets.  This definition is the first my PhD supervisor showed me in my very first supervision.

You can think of building Schreier sets by picking the smallest element, n say, and then filling the set up with at most (n-1) strictly greater distinct numbers.

The Schreier family has the following simple properties (that make it useful when defining norms on Banach spaces, especially the Schreier space):

  • The singleton \{n\} is Schreier set for all n \in \mathbb{N}
  • The family is hereditary: each (non-empty) subset of a Schreier set is itself a Schreier set.
  • The Schreier family is spreading: if \{m_1, m_2, \ldots m_k\} is a Schreier set and n_j \geq m_j for all 1 \leq j \leq k, then \{n_1, n_2,\ldots, n_k\} is also a Schreier set. (These don’t necessarily need to be in ascending order).

Schreier and Fibonacci

Many of areas of maths have links to the Fibonacci sequence: here’s one, involving the Schreier sets, that I haven’t seen before.

Define M_n to be all those Schreier sets whose greatest element is n. For instance M_5 = \{ \{5\}, \{5,2\}, \{5,3\}, \{5,4\}, \{5,4,3\} \}. Then the number of elements |M_n| = F_n the n-th Fibonacci number.

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Jenga blocks and Gowers’ hyperplane space

This post is an attempt to communicate some of the feel of Banach space theory to those who aren’t familiar with it. I once tried to explain my research to a six year old using Jenga blocks, but fortunately only got as far as the triangle inequality. Near the end of my Phd, at my supervisor’s suggestion, I started to explore the complicated Banach space that is Timothy Gowers’ solution to Banach’s hyperplane problem. These experiences inspired the following explanation of one relatively simple observation (that I included as an example in my thesis) through the delightful medium of building blocks.

Our object of study are towers of good old-fashioned building blocks. Each block has a number written on its side, so each tower built from these blocks gives a sequences of numbers (x_1, x_2, x_3, \ldots). These don’t have to be positive natural numbers, but you won’t lose much by pretending, in this post, that they are. There are innumerably many different brands of towers, but we’ll concentrate on one particular brand: the ‘Gowers Towers’. Let’s say the number written on each block represents how heavy the block is, and is inversely proportional to the length of the block. So we’d represent the sequence (0, 1, 5, 0, 3, 5, 5, 1, 10) with the Gowers Tower pictured.

It’s worth mentioning that the Gowers Towers include every individual tower of finite height that you can build with your unlimited set of Gowers branded building blocks (and lots of infinite height, but you don’t really need to worry about those here).

Let’s pretend we’ve got a measure of the instability of a tower (the norm of the sequence), and whenever we increase the instability beyond a certain threshold, K, the tower collapses.

Blocks with higher numbers are heavier, as well as narrower and perhaps inherently more unstable. How the blocks of different weights at different heights affect the stability of the Towers of Gowers is extremely complicated. However, the towers do have some nice, intuitive properties.

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Getting into Norms: Part II

In Part I of Getting into Norms, I talked about three different ways of measuring distance (I also considered  the accuracy of a series of guesses to be a ‘distance’). All three of these were norms, but there are many ways of measuring distances that aren’t norms.

So to study norms, mathematicians must define them really rigourously, using something known as axioms. These are the basic assumptions and definitions of mathematics. Once we’ve made these assumptions we can prove what has to follow from them.

We can think of norms as a measure of distance from the origin. If you think about it in this way, the following seem quite obvious, and appeal well to our instincts. A norm satisfies the following three axioms.

  1. Distances are always positive!
  2. If the distance from your location to the origin is zero, then you must be at the origin. Or alternatively, if two points are separate then the distance between them isn’t zero. Conversely, the distance from any point to itself is zero.
  3. Taking a detour is always longer than travelling in a straight line. This is the triangle inequality: the sum of the length of any two sides of a triangle is longer than the length of the third.
  4. Now we come to axiom four. This one is tough to describe in words. Here goes. If you walk a pace forwards and then take another in the same direction, then you will have walked twice the distance of the original pace. Also it doesn’t matter whether you take a pace forwards or backwards: they will give you the same distance.

When mathematicians want to be precise, we use symbols. The distance between points x and y is written as \|x-y\|. The distance from x to the origin is \|x-\underline{0}\|=\|x\|. We say that \| \cdot \| is a norm if whenever we pick vectors x and y, and a number \lambda, then the following axioms hold:

  1. \|x\| \geq 0.
  2. If \|x\| = 0 then x = \underline{0}. And visa-versa.
  3. \|x+y\| \leq \|x\|+\|y\|.
  4. \|\lambda \cdot x\| = |\lambda| \|x\|.

These four conditions should match with our verbal descriptions above. You may recognise them from this blog’s exquisitely hand-drawn logo.

They were pretty trivial intuitions, once we thought of \|x\| as being the distance of a point x from the origin (the origin above \underline{0} is underlined to distinguish it from the normal 0, though we don’t choose a different notation because the origin behaves a lot like the number zero). Continue reading

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