Package 3

Puzzles

Resources

Introductions to Mathematics at University:

"Advanced Problems in Mathematics: Preparing for University" by Stephen Siklos free pdf book
NRICH collection to prepare for University
Problem solving strategies with examples
Videos at 3Blue1Brown on many advanced maths topics

Pandemics, Mobile Apps and the Maths:

Currently frequently in the news is the introduction of mobile phone apps for contact tracing while opening up the country. The goal is to decrease the average infection rate R0, ideally to get it under 1, which means that infections will decrease.

In 2018, "Contagion: The BBC Four Pandemic" (also available here) conducted a nationwide citizen science experiment led by Hannah Fry to help plan for the next deadly flu pandemic, which could happen at any time. More background about the experiment, the app and the contributors.
Hannah Fry's podcast from 2020 reflecting on BBC's Contagion.
Interview with Julia Gog (Professor for mathematical biology in Cambridge) about making Contagion.
The mathematical theory underlying Contagion: Effectiveness of isolation, testing, contact tracing and social distancing of SARS COV-19 in different setting.
Christoph Fraser and his team at BDI have been working on the concept and implementation of phone app for tracing Covid-19 infections based on their publication Quantifying SARS-CoV-2 transmission suggests epidemic control with digital contact tracing.
Michelle Kendall's R shiny app provides simulation tools to create infection and intervention scenarios.
Why It's Not Too Late for Contact Tracing and other topics at Podcast-19 by FiveThirtyEight

Mathematical knowledge, beauty, curiosities, puzzles and more:

Chalkdust is a magazine for the mathematically curious. It is a collection of pieces of paper covered in interesting content and stapled together. Check out their current issue and their archive with previous issues. Under How to make it they also explain how to surprise Möbius, how to bake tessalation shortbread and how to make a catastrophe machine.
Probability puzzles by puzzlewocky and by Braingly, and two children from Alex Bello's Monday puzzles.

Lying with Statistics:

Disraeli's famous quote "There are three kinds of lies: lies, damned lies, and statistics." and Darrel Huff's often quote book "How to lie with Statistics" with detailed instructions on how to do so do not give our field a good press. However, such lies are typically not produced by statisticians, but by people trying to push an agenda (or inadvertently). The statisticians' role is in supporting citizens in spotting such lies. Here is a list with the most basic ways to cheat, so you get experienced and catch the liars quickly:

TedEd talk and resources by Lea Gaslowitz about How to spot a misleading graph.
Blunt example: a recent chart by the State of Georgia and the corrected version.
Collection of misleading graphs and statistics with discussions.
Wikipedia page "Misleading graph"
Spurious correlations shows how mozzarella consumption is related with Engineering doctorates and many other entertaining coincidences.

Visualisation and simulations of stock price data and financial instruments

Have you ever plotted stock price data? This type of data is very easily to download from the web. It has very different characteristics from the ecological or infectious disease data we saw in the two previous packages. For example, measurement error and missing data are less relevant. Usually, no information on a battery of related phenomena come with the data. The exception are other stock's prices, and analysing associations between stocks and portfolio level investment decisions making is a research area in both conventional and behavioural finance. Here are some starting points for getting a very first taste of computational finance.

PhD student Curtis Miller's introduction to stock market data, on how to get it and visualise it in R. There is also a Python version of this. (In case you get hungry from all this, the same author also provides donuts.)
Here is some sample R code to simulate stock price data. It creates a number of realisations of the same kind of stock (i.e. based on the same stochastic model). If you run it repeatedly you get another bunch of realisations. Not surprisingly, they are all different.
The R package derivmkts contains a lot of more sophisticated R code that you may explore for simulating stock market data and for experimenting with financial derivatives. Scroll down on the GitHub page more an overview of the package. Check out the Readme files and the Vignette to learn what this package can be used for in more detail. The functionalities include simulations based on the binomial model as well as implementations of financial derivatives theory (e.g. Black and Scholes formula based hedging). This is a lot to take in, but if you are interested you may want to first acquire some knowledge about the financial concepts, terminology, models and strategies:
- Trader slang for some parameters to do with characteristics of stocks is "the Greeks", because whoever introduced them that was a really cool name and now we are stuck with it.
- Now it is time core finance, stocks & bonds and options (various types of these and pricing based on hedging strategies such Black & Scholes) kindly provided by Salman Kahn, who grew up Bengali-American and later ditched an investment banking job for creating an amazing online learning platform. (How this happened has been told at Rice University Commencement speech. Note, true to these roots he created homeschooling resources within days of the shelter-in-place order in the US to support children and young people in continuing their education despite Covid-19 school closures. I your younger sibling during UK lockdown have enough of MyMaths, that may be the answer for them as well.)
- More about Black and Scholes formula by and about the construction of a binomial tree to describe stock prices movements by Ted Stephenson.

Remember, you can find resources on how to learn R in Package 1. If you know another languages, e.g. Python, you may use this as well.