I was interviewed by Nira Chamberlain, President of the Mathematical Association. I am the twelfth person to whom he has asked his question “what is the point of mathematics?” Hoping to offer something a little different, I spoke about teaching students the role mathematical modelling can play in sustainability.
You're reading: Travels in a Mathematical World
A visit to The Mathematikum in Giessen
My son and I visited The Mathematikum in Giessen. This is well worth a visit, we did it as a day trip by train from holiday in Frankfurt, which worked well because the museum is close to the railway station. The Mathematikum specialises in ‘hands on, minds on’ interactive activities, and we spent about 5 hours exploring the four floors. I enjoyed the open-access article The Mathematikum in Giessen by Martin Buhmann, who was kind enough to meet us and show us around.
There are some Mathematikum-made exhibits at MathsCity Leeds. I took some pictures of exhibits we had enjoyed that aren’t (to the best of my memory) available in Leeds. Here they are, in no particular order.
Bouton numbers: a new integer sequence
In the 1901 paper that named the game Nim and provided its mathematical analysis, Charles Bouton defined “safe combinations”, positions that if you leave the game in this state, your opponent cannot win. In combinatorial game theory, these are \(\mathcal{P}\) positions (the previous player has already won), as opposed to \(\mathcal{N}\) positions (the next player can win).
Bouton gives a list of “the 35 safe combinations all of whose piles are less than 16”, working in three-heap Nim. Naturally it seemed sensible to check these, so I wrote a bit of Python code to do this. Bouton’s list is good. I realised I could easily adapt my code to find out how many \(\mathcal{P}\) positions there are for three-heap Nim games with other maximum heap sizes: 1, 2, 3, and so on.
And, having generated a sequence of integers, I naturally looked to see if it was in the OEIS. This is sometimes a good way to discover that your sequence of numbers is also found in some unexpected places. It wasn’t there! So I submitted it, and I just got the exciting email “N. J. A. Sloane published your changes”. So I present A363166: “Bouton numbers: a(n) is the number of P positions in games of Nim with three nonzero heaps each containing at most n sticks”.
This is my first OEIS submission, so it’s all very pleasing, even if I’m submitting a ‘new’ sequence inspired by a 1901 paper!
27 tickets that guarantee a win on the UK National Lottery – but what prize?
The recent preprint ‘You need 27 tickets to guarantee a win on the UK National Lottery‘ by David Cushing and David I. Stewart presents a list of 27 lottery tickets which will guarantee to match at least two numbers on the UK National Lottery, along with a proof that this is the minimum number you need to buy. The argument is clever and makes delightful use of the Fano plane.
I wrote some Python code that runs all 45,057,474 possible draws against these 27 tickets.
All draws had between 1 and 9 winning tickets from the set (crucially, none had zero!). Obviously for 27 of the draws one of the winning tickets matched all six numbers, but about 75% of the draws saw a maximum of 2 balls matched by the winning tickets, and a further 23.5% had at most 3 balls matched. This means almost 99% of the time the 27 tickets match just two or three balls, earning prizes which may not exceed the cost of the 27 tickets! (I recommend reading Remark 1.2 in the paper.)
More findings and my code on GitHub.
Update 1: Tom Briggs asked what’s the expected return for buying these 27 tickets. I think the average return is about £20, which is a £34 loss (and of course this is an average from a set of numbers that includes some big wins). Assumptions and details in the GitHub.
Update 2: Matt Parker prompted me to investigate what percentage of draws end in profit. Even though 99% of the time the tickets match just two or three balls, if more than one ticket matches three balls that would still be a small profit. In fact, a profit is returned in 5% of draws, though as noted above the expected return is a loss. Matt included this result in a fun video about the 27 tickets. Again, assumptions and details in the GitHub.
Shoes and partition numbers: a developing mathematical mind wanders
Does this picture make you think of Srinvasa Ramanujan? I’m always fascinated by the pace and range of little conversations with my seven-year-old son that wander in and out of maths. Let me tell you how we got there during a five minute chat while leaving the house and walking to school this morning.
Twitter and Mathstodon
Some thinking aloud about what’s happening on social media in my world, I hope you don’t mind.
customdice: a new LaTeX package for drawing dice
I made a new LaTeX package for drawing dice, customdice.