Open Mic A

 

James Bromby: The Problem with a BIG MAC for a Happy BirthdaY

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 Vavasour: PiE Base

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

 

Rata Ingram: How To Subtract with an Adding Machine

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 Eddy: What Approach?

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 Macleod: Mathemagick

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...

 

Ross Atkins: Mathemagick 2

The mathematical magic tricks continued with a game of shuffling forks and some suspect matrix multiplication. Can you see what is wrong the the matrix multiplication below?: 

Open Mic B

 

Kirk Alexander: Mutterings About Enigma

A WWII-era Enigma machine consists of an electrical circuit linking a key from a keyboard for an alphabet of A characters through a state machine to one of A lamps. The machinery providing the variable circuit is a single fixed reflector (chosen from a set of R different possibilities), a set of n rotors (chosen without replacement from a bag of N possibilities) and a plugboard of A sockets, into which q wires connecting two sockets each are plugged.

The presence of the reflector means that the same configuration of the machine can both encrypt and decrypt, vastly simplifying setup and training. For example, if the current setup will encrypt ‘B’ to ‘T’, pressing the ‘T’ key instead will result in the ‘B’ lamp lighting up.

 

Kirk gave us a crash course in how such an Enigma machine works, explaining how each key part contributed to the goal to write an encrypted message. He then calculated the number of different states that could be achieved using different combinations & setups of both the rotors, and the plugs in its plugboard. 

 

Noting that the rotors by themselves provide relatively small protection against a brute force attack - their main effect was the non-linearity of the wirings making the scramble unpredictable from one step to the next - he showed with an actual used German configuration (R = 1, N = n = 3) the resulting number of rotor states S, is 105456.

 

• In contrast the number of plugboard setups is A! / { 2^q . q! . (A – 2q)! }. Using the typical setup of 10 wires in the Enigma's plugboard gave a phenomenal value of about 15 x 10¹³ states, providing extensive protection against a brute force decryption attack, despite the simple effect of each plugboard wire on the characters. Kirk showed that the optimal value however is q = 11 wires, leaving the question open on why Germany didn't use this figure instead.

 

Scott Wilson: Triangles

Triangles are everywhere, let us count the ways:

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 Witham: Mathematics of Poetry in Ancient India

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.

 

Merlyn Barrer: Origami Folding

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

 

Andre Macleod: Mathemagick Rebooted

This time, the trick went without a hitch.

 

Open Mic C

 

Louis Ammundsen: .*(D|R)NA.*

Nucleotide sequence alignment underlies all the revolutionary technological development since the development of DNA sequencing. But the NCBI databases include hundreds of millions of sequences, comprising hundreds of billions of base pairs, so how do we search this database to study samples? And how do you test the significance of matches, especially when they are imperfect? Louis' talk gave a quick overview of the BLAST algorithm and its applications.

 

Maia Traforti: Pure Math for Kids

Maia outlined some of her ideas for engaging kids of all ages with mathematical thinking from an educational perspective.

 

Elizabeth Chesney: From Mother Nature to MAthematics

Elizabeth and David presented a powerpoint slideshow featuring the many and varied places one can find fractals, how they form and what they represent mathematically, touching on Mandelbrot and the Julia set. The presentation was originally written for SeniorNet Mac Inc., a community training / learning network for seniors using computers.

Download the slides here [PDF] or view as Google slides here

 

Todd Rangiwhetu: Visualising Mathematics

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 Andrew: Improving Maths Education

 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 Ingram & Amy Reddecliffe: Mathateca

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.

1. MathsJam brownie - Sophie
2. Golden Madiera cake - Tom
3. Mathsghan biscuits - Rata
4. Square pyramidal spinach ball towers - Rata
5. Towers of Hanoi - Josie
6. Cinnamon scone Möbius strips - Andre

 

Winner best mathematics: Andre (6)

Winner best tasting: Sophie (1)

Winner best looking: Josie (5)