Join the Fitzwilliam Maths Circle
A new outlet for learning together
If you are reading this post, you will almost certainly be interested in the update about our efforts to help the Irish Mathematics Olympiad. Thank you all for your support!
The Fitzwilliam remains entirely free. However, Sam Enright’s Newsletter has now opened up paid subscriptions. Paid subscribers will support my work, have access to any subscriber-only bonus posts, and have priority for attending any space-limited events. We have disabled the ability to ‘pledge’ to support The Fitzwilliam, and encourage anyone who would have paid for a Fitzwilliam subscription to subscribe to my newsletter instead.
We have been very pleased with how The Fitzwilliam Reading Group has been going. This month, we talked with Noah Smith about whether data centres are going to crash the economy. Next month, we’ll be talking with Gavin Leech about his new book with Dwarkesh Patel about recent progress in AI. In the new year, we will be discussing the poetry of Seamus Heaney.
However, there is only so much you can learn from just sitting around and talking. Indeed, one of the areas that struggles the most with reading groups is something that many of us wish we did more of: mathematics.
And so, The Fitzwilliam is starting a monthly maths meetup.1 The idea is that we will distribute problems before each meetup, everyone will try them, get stuck, and then we will gather together to learn in a fun and supportive environment.2 There’s no specific number of problems you have to have completed before the meeting; it’s more about trying your best to figure things out yourself before seeking help.
No one from The Fitzwilliam studied maths at university. As with almost everything I do, I am woefully underqualified for this. What I lack in experience, I make up for with infectious joie de vivre.
And to be honest, most of the heavy lifting here is being done by my good friend Neil Shevlin, who has recently joined the ranks of funemployment.
My vision here is that telling someone that you’ve never done maths for fun will be like saying that you’ve never listened to music or watched a film. While I would not dare to underestimate the lottery of fascinations, if you never do maths, there is so much of the human aesthetic experience that you are sealing yourself off from.
Our first meetup will take place on Saturday, November 29th, at 3pm. It will be in the Innovation Space of Dogpatch Labs, immediately adjacent to the Starbucks. It will be on the topic of induction, and you can download the problem set here.
The material is anchored by chapter 4 of Proofs: A Long-Form Mathematics Textbook by Jay Cummings. I strongly recommend getting a copy of the book and reading along. Unlike most maths textbooks, it contains actual sentences and paragraphs, which is a necessary condition for my wordcel brain to understand.
The second meetup will take place on Saturday, December 13th, also at 3pm in the Innovation Space. The topic is number theory, and you can download the problem set here. This is inspired by chapter 5 of Math History: A Long-Form Mathematics Textbook, and you can follow along with the relevant lecture slides from the author’s website.3 I would also recommend ordering a copy of the book.
We hope to have provided enough resources so that people with a wide range of abilities will be able to attend.
If any of this sounds appealing to you, please fill out the form to join here. [Edit: I ported this form over on 25/11/25. If you are reading this more recently and would like to join, please send an email to sam [at] thefitzwilliam [dot] com with a brief bio.] As with the reading group, there will be announcement emails, and a WhatsApp group for chatter and sharing puzzles. You are welcome to sign up even if you have a low probability of being in Dublin on any given month.
The exact format will probably evolve over time. While reading groups are a tried-and-tested format, this is a bit more experimental, and you’ll have to be patient with us. I don’t know enough to say how similar this may or may not be to the historical phenomenon of “maths circles”.
While reading groups are somewhat common, meetups to learn mathematics as an adult not in university seem to be almost non-existent. In stark contrast to Ireland, when you visit the Bay Area, it’s not unusual for offices to have recreational mathematics groups that meet on evenings and weekends. Despite my significant misgivings, I admire how the people there pursue inherently challenging hobbies and side interests with utmost sincerity and dedication. I think that is one of the reasons why San Francisco is Mecca for nerdy Irish youngsters, and why emigrating there in connection with Patch is the Hajj. I am not entirely sure where our group will fit into this Islamic pilgrimage analogy, but we hope it will be a worthwhile contribution. Please join!
Sam Enright is editor-in-chief of The Fitzwilliam, and Innovation Policy Lead at Progress Ireland. You can follow him on Twitter here or read his personal blog here.
At first, our plan was to start a MathsJam in Dublin. MathsJam is a series of monthly meetups in pubs around the world for “maths enthusiasts” to get together and share puzzles, games, and miscellany with each other. Despite the fact that there is a page on the website for the UK “and Ireland”, there is no Irish meetup. Eventually, we decided that MathsJam was not the right outlet for what we had in mind, but if you would like to start a MathsJam of your own, and we can be of help, please email sam [at] thefitzwilliam [dot] com.
While I was writing this post, a friend messaged to ask “Should I ask the 10/10 baddie out on a date to watch a film or do functional analysis problems :??”, as if he didn’t already know the answer.
For November’s meetup, some of the problems were taken from chapter 5 of Discrete Mathematics and Its Applications by Kenneth Rosen. December’s meetup is partly inspired by chapter 4.
I had a quick look through the problem set and since the topic was induction I thought it would be fun to think through how you'd do all of the problems directly (without induction).
The most fun one I think is how you compute 1^2 + 2^2 + ... + n^2 without induction.
I've seen the answer and inductive arguments lots of times but have never thought about or seen how you do it directly, and so I think I've rediscovered the combinatorial way to do it.
---
We want to compute 1^2 + 2^2 + ... + n^2.
What does each term k^2 count? Fix a number k. We can think of k^2 as the number of ways to choose an ordered pair (a, b) where each of a, b is an integer from {1, 2, ..., k}.
This means the whole sum counts the number of triples (k, a, b) where 1 <= a, b <= k <= n. Now we build all such triples in a different way.
The trick is to work inside the set {1, 2, ..., n+1}, and let k+1 always be the largest number we pick. That way, anything else we pick is automatically at most k, so it’s a valid choice for a and b.
We split into two cases:
1. Triples with a != b: Choose 3 distinct numbers from {1, ..., n+1}. Call them x < y < z. Let k = z − 1. Then (a, b) can be (x, y) or (y, x). Both choices have a, b <= k, so both give valid triples (k, a, b). So each 3-element subset gives 2 different triples with a != b.
2. Triples with a = b: Choose 2 distinct numbers from {1, ..., n+1}. Call them y < z. Set k = z - 1 as before, and a = b = y. Then we get a valid triple, and so each 2-element subset gives exactly 1 such triple.
These two processes produce all triples (k, a, b) with 1 <= a, b <= k <= n, and nothing is missed or double-counted So we can count them:
There are C(n+1, 3) ways to choose 3 distinct numbers; each gives 2 triples.
There are C(n+1, 2) ways to choose 2 distinct numbers; each gives 1 triple.
Therefore,
1^2 + 2^2 + ... + n^2
= 2 * C(n+1, 3) + C(n+1, 2)
= 2 * [(n+1)n(n-1)/6] + [(n+1)n/2]
= (n+1)n(2n+1)/6.