Thanks to the wild and windy weather on Saturday, our venue unfortunately lost power for several hours during the time the keynote was scheduled. Todd was able to improvise and modified and delivered his talk off the cuff, with no slides.

Abstract: Let's make full use of our pattern-recognition skills, and take a journey through some visualisations of mathematical ideas and constructs. We'll see some of the free tools currently available (such as Geogebra) and find out what we can do with them, as well as revisiting some interesting historical maths visualisation tools. This will be a fun, demonstration-based talk.

**Todd Rangiwhetu** is a human who loves maths. That has manifested in many forms, including turning coffee into

theorems, teaching in NZ and the UK, writing online and TV mathematical content, and MathsJamming since it started in a pub in London. Todd came all the way from Wellington, where he co-organises Welly MathsJam.

*Open Mic Maths is a series of ~5 minute, informal presentations about anything maths related; all attendees at OMG are encouraged to speak in one of the sessions. Each scheduled block was followed by a breakout session for further exploration of the topics covered. Read the talk summaries provided by attendees for OMG23 below.*

Computers communicate with one another using IP addresses (IPv4 & IPv6) and MAC addresses. For privacy reasons a MAC address can now be locally administered - which means it will be (pseudo)randomised. So how do we ensure two devices on the same network don't happen to locally administer the same address?

James compared this with the similar Birthday Problem, which asks: in a room of N people, how likely is it to have two who have the same birthday? He showed that the number of possible MAC addresses is huge compared to the number of days in a year, so using the same formula pushes the values beyond the capability of a pocket calculator! Instead, a Taylor Series expansion can be used to approximate the answer.

Related link: BAWMAN - Random MAC

Tom is back to talk about pi from another angle - this time exploring what a system of numbers would look like in base pi.

Adding machines are simple devices that allow you to add up numbers by pressing buttons that progress a rachet counter mechanism by the corresponding amount. But what if you've added too many figures? What if you'd rather take a number away? Can it be done on an adding machine, or is subtraction the device's antithesis? Do we need to go back to the future and the comfort of our four-function pocket calculators, with their trusty minus buttons?

It turns out we can subtract on an adding machine, using the "nines complement" trick. Find the nines complement of every digit you want to subtract (including the leading zeroes) and add the resulting number to your total on the machine. Then add one, and disregard the leading one.

In her talk, Rata showed why the trick always works, and even brought along a couple of old adding machines to demonstrate in real time.

David has observed that people tend to have two distinct approaches to solving the Tennis Tournament problem. He was curious to see which way the audience went, so posed the problem and polled us on our approaches.

Read about the problem here: Tennis Tournament Problem

Andre talked us through how to perform a card trick that finds a Jack in a shuffled deck. He also showed us how to play off a trick when it doesn't quite act as expected...

The mathematical magic tricks continued with a game of shuffling forks and some suspect matrix multiplication.

How does the Enigma machine work?

Triangles are everywhere, let us count the ways:

**Triangle Subtraction (TriangleSub.gif)**:

Take a triangle (equilateral for simplicity). Remove a half-sized triangle from the centre; now there are 3 triangles. Do the same centre-removal to those triangles; now there are 9. Repeat ad infinitum & get the Sierpinski gasket.

**3 Points (Points.gif):**

Start with 3 points in a triangle. Put a random starting point within the triangle. Pick a corner at random & put a point halfway between the corner & the new point. Repeat ad infinitum, using each successive point & a random corner, to get the Sierpinski gasket.

**Pascal's Triangle (Pascal.gif):**

Pascal's Triangle (each entry the sum of the 2 above) has many strange properties. One interesting pattern emerges if you work mod-2 (or more simply, looking at "odd or even"); a Sierpinski gasket. After setting this up in a spreadsheet & using colouring to show the result, I wondered about other "mods". Mod-3 & mod-5 are shown; an obvious pattern related to the base can be seen.

**Conway's Game of Life:**

Conway's well-known 2-D Game of Life has a 1-D equivalent (no or 2 neighbours = dies, 1 neighbour = lives). By then using "time" as a vertical dimension for ease of display, it becomes 2-D again but with slightly different rules. Starting with a single point, as time progresses (traditionally filling rows below the previous) the Sierpinski gasket will emerge. (The logic behind it is identical to "Pascal's Triangle mod-2".)

**Collage Theorem:**

Take an image (doesn't actually matter what it is...). Replace it with 3 half-sized copies, in a triangular arrangement (touching butnot overlapping). Using this "whole image", repeat "copy, shrink, replace" ad infinitum to create the Siepinski gasket. (The Collage Theorem is a very powerful iterative method for creating all sorts of imagery.)

Sophia has been studying poetic forms and meter from Ancient India. There is a lot of mathematics buried within, including the Fibonacci numbers, and the concept of zero. Based on her research:

Montelle C, Witham S. (June, 2023) *Historical Notes: Finding Mathematics in Poetry, * pp. 212-213. Mathematics Today.

Modern origami artists use some fairly elaborate techniques to construct intricate and realistic designs. Merlyn provided an overview of how he, as an origami artist, creates the crease pattern for a new origami design, conceptualising the desired shape as a stick figure and using a disc packing technique.

Check out Merlyn's work here: https://www.instagram.com/merlynorigami

This time, the trick went without a hitch.

TC

Maths education - ideas for engaging kids of all ages with mathematical thinking

Fractal talk

With the power back on, we got a quick version of the slides to complement Todd's keynote talk, including some neat Geogebra demonstrations.

Richard is the driving force behind Learn Implement Share. He cares about getting mathematics teaching right and is not afraid to challenge the status quo. In his talk, he explored the problem of "The Memory Game" – a term he coined to describe the school experience of students who lack a conceptual understanding of much of the work they undertake in lessons, with the open question "How do we improve mathematics education?" left for an invigorating breakout chat.

Read on LinkedIn: It's Time To Stop Our Mathematics Students From Playing The Memory Game!

Rata spoke about the idea behind, and the founding of the organisation Mathateca, which was launched earlier in 2023 with Hex day at Christchurch City Libraries. She also highlighted the upcoming event Show Your Working organised by Amy for November 2023.

*Mathateca is a movement based in Christchurch, New Zealand, which aims to create a public space dedicated to mathematics so that everyone has the opportunity to engage with maths recreationally. Just as art galleries do for art, museums for history, and libraries for stories, Mathateca strives for informed, creative maths curation in order to celebrate mathematical ideas and learn in the community.*

- MathsJam brownie - Sophie
- Golden Madiera cake - Tom
- Mathsghan biscuits - Rata
- Square pyramidal spinach ball towers - Rata
- Towers of Hanoi - Josie
- Cinnamon scone Möbius strips - Andre

Winner best mathematics: Andre (6)

Winner best tasting: Sophie (1)

Winner best looking: Josie (5)

Origami Number competition - Merlyn

Prime Loop competition - Kirk

Number Card Game - Rata

Letter Meta Puzzle competition - Ross

Puzzle Vision Competition - Rata

How many sweets in the jar? - Josie

Totally genuine and not contrived competition - Todd

Winner people’s choice: Josie (Sweets in the jar)

Winners chosen by judging panel: Kirk (Prime Loop), Rata (Puzzle Vision)

*NB. No subversion was detected by our judging panel so two judge's choice were awarded*