The following problem popped into my head this morning on the one hour drive home to Gympie from Rainbow Beach. In the oncoming traffic, cars often appeared in clusters. Indeed, if there is one car that is going more slowly than the others, then they are all pretty much stuck behind it (unless they overtake). The problem I’m thinking of is this: what is the expected number of cars in each cluster? Like most problems that just pop into my head, this one has likely been thought of and solved many years ago. It’s also not very well-formed. So let’s develop a simpler statement and have a think abo ..read more

The YouTube channel Mathologer posted a wonderful demonstration of a classical result, I thought I would give my own twist on it ..read more

Are you a student in graph theory, design theory, finite geometry, or any other area of combinatorics? Do you use combinatorics in your research? If yes, we invite you to attend the Student Symposium in Combinatorics 2022. This online symposium is for students by students and is part of a collection of student symposia supported by MATRIX. SSC is a welcoming and friendly space for students of all stages (PhD, Masters, Honours, Undergraduate etc.) to discuss everything and anything related to combinatorics. We have a great lineup of invited student speakers from around the world, inc ..read more

Here’s a good one straight out of Martin Gardner’s Colossal Book of Short Puzzles and Problems. My wife and I recently attended a party at which there were four other married couples. Various handshakes took place. No one shook hands with himself (or herself) or with his (or her) spouse, and no one shook hands with the same person more than once. After all the handshakes were over, I asked each person, including my wife, how many hands he (or she) had shaken. To my surprise each gave a different answer. How many hands did my wife shake? Solution to come soon! The post Problem: How many hands ..read more

The post Proof Without Words – Recursive Squares first appeared on Dynamical Mathematical Gazette ..read more

You and your friend stand at opposite corners of an grid. You are at the bottom left; your friend is at the top right. Every second, you will randomly (according to a coin flip) move either right one space or up one space (though if you can’t move in one of these directions because you are at that edge of the grid then you can only proceed in the other). At the same time, your friend will randomly move either left one space or down one space. Problem. What is the probability that you and your friend will meet at some point in time? Let’s solve an easier problem together and then you can do th ..read more

A medley of drums. Following from my previous post, here’s another surprising crossover between maths and music. Did you know that some composers use number theory to write music? We can think of a rhythm as notes distributed in a measure of fixed length. We’ll ignore syncopation for the sake of simplicity. For example, in a measure of 4/4 time, there are four positions in which a crotchet could be written (or a quarter note for my fellow Americans). I claim it’s possible to determine a rhythm by using the Euclidean algorithm. The main idea is that equally distributing notes in a measure is eq ..read more

I write music in my spare time. It’s fascinating how often mathematical ideas come up in that domain. One of my side projects this year is to learn some algorithmic composition techniques and implement them in code. In my mind, this goes back to the 20th century, with Arnold Schoenberg writing out permutations of his tone rows. So imagine my surprise when I stumbled on a permutation algorithm that’s hundreds of years older! I’m talking about traditional church bell ringing. The problem is straightforward. You have a fixed set of bells, commonly steeple bells or handbells, which ring in sequen ..read more

A marked lotto ticket. Here’s a chance to flex those combinatorics muscles. A lottery is drawn by selecting five integers at random from 1 to 69 inclusive, without replacement. The result is reported in increasing order. What is the probability that the pair “20 22” occurs consecutively on the winning ticket? To be clear, “05 10 19 20 22” counts, but “05 10 20 21 22” does not. Once you’ve got that worked out, find a lucky year; one for which the first two significant figures of the probability, expressed as a percentage, are identical to the year modulo 100. For example, 2029 is a near miss wi ..read more

The Student Symposium in Combinatorics is happening next week and they are running a couple of online panel sessions that all students of maths might find pretty useful! You can find out more and register for the panels here. I’ll be a panellist for Panel 1 and will have some useful things to say about using your maths toolkit to chisel your way into industry. The post Upcoming SSC Maths Panels first appeared on Dynamical Mathematical Gazette ..read more