### SINGLE USE PLASTICS FROM ONE TO A TRILLION –

### GET READY FOR SOME HUGE HUGE NUMBERS!

**ONE:**

This is a one litre bottle of water.

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## A Mathemagical Sleight of Hand:

Last night saw the live final of TV Talent Show Britain’s Got Talent. The evening’s second most popular act “X” was a masked magician, presumably named “X” in homage to his love of the algebraic symbol used to represent the unknown. Viewers of a mathematical persuasion will have noticed that runner-up “X”‘s big magic trick of the evening used not sleight of hand, but rather sleight of mathematics. The voting general public were impressed enough to make X the evening’s runner-up, losing out only to octogenarian singer Colin Thackeray.

## The magical effect:

Here’s a photo of the act in progress:

The mathemagician can now demonstrate “mind-control” as follows:

Start at any circle

Move left or right to the nearest square.

Move up or down to the nearest circle.

Move diagonally to the nearest square.

Move left or right to the nearest circle.

Are you now on HOPE? Yes, I thought so!

I’ll not spoil your fun – see if you can work out for yourself how this trick works!

# Star Polygons: an introduction

Star polygons are beautiful mathematical objects, a juxtaposition of maths and art. They’re also really easy to make: as an example here is a {5,2} Star Polygon. The first number tells us to start with 5 points (“**vertices**” or in the singular “**vertex**“) in a circle. It’s convenient to number them 0, 1, 2, 3, 4. (You can label them 1, 2, 3, 4, 5 if you prefer). The second number tells us which points to connect: you count on 2 places each time. So start at 0, join to 2, then 4, then 1, 3 and finally back to 0: the star is complete! Note that arithmetic behaves quite unusually when creating a star polygon: for example 4+2=1. Mathematicians call this **modular arithmetic** and it crops up all over the place, most obviously on clocks (e.g. 10 o’clock + 5 hours = 3 o’clock!!).

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__DO’S AND DON’TS WITH A COMPASS__

## LEVEL: UP TO GCSE

**1. “A PAIR OF COMPASSES”:** does not mean you should use two of them! Same comment applies to wearing a pairs of trousers or glasses.

**2. USE A TEENY WEENY PENCIL:** Long pencils get stuck against your hand and prevent the compass moving properly, but a tiny stubby pencil can stay in the compass without getting entangled in your pencil case. If you snap a small piece off the end of another pencil and resharpen the end your Mum will probably not notice. I’ve been using the same compass pencil for years, and it still has plenty of length left.

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__A SURPRISING FORMULA FOR GENERATING PYTHAGOREAN TRIPLES__

A Pythagorean Triple is a set of three positive integers (whole numbers) that satisfy Pythagoras’ Theorem $a^2+b^2=c^2$, such as {3, 4, 5} or {5, 12, 13} or {28, 45, 53}. It’s easy to see that there are infinitely many such triples: one way is to take multiples of the well-known {3, 4, 5} triple; multiplying each number by 2 or 3 etc we find that {6, 8, 10} or {9, 12, 15} etc also satisfy $a^2+b^2=c^2$. A more interesting way to generate infinitely many Pythagorean Triples is as follows:

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# DISSECTIONS OF AN OCTAGON

**OCTAGON-INTO-QUADRILATERALS**

**WHAT FRACTION IS SHADED RED?**

“Geometry” is a posh word for “shapes”. Here are two fun geometry puzzles for you, inspired by a question form the UK Mathematics Trust Challenge. Anyone can play, even non-mathematicians:

- The first octagon has been cut into four
**congruent** (same size, same shape) quadrilaterals (four sides). Can you figure a way to dissect the octagon into four congruent **pentagons **(five sides)? How about into four congruent **hexagons **(six sides) – or even heptagons and beyond?
- What fraction of the red-and-white octagon has been shaded red? (and can you prove it?)

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__Or: a beginner’s guide and glossary for the 4__^{th} spatial dimension.

**INTRODUCTION**: imagine a** tightrope walker**: she can only change her position in one direction: forwards & backwards, so we need just one number – how far along the rope she is – to specify her position. That’s **one dimension**!

Now picture an **ant crawling on a tabletop**: the ant can crawl forwards & backwards *or* left & right, and we now need two numbers (e.g. the ant’s horizontal and vertical distance from a particular corner of the table) to pinpoint exactly where the ant is. That’s an extra degree of freedom: **two dimensions**!

And finally, picture **yourself wearing a jetpack**: you can now move freely in all three of our spatial dimensions: forwards & back, left & right, AND up & down. Three numbers are now required to describe your position at a given time e.g. latitude, longitude and altitude. That’s **three dimensions**!

If you think that sounds fun, imagine how cool it would be if there was a fourth, new direction in which you could also travel! What would **4-dimensional space** look like, and what sort of shapes would inhabit it?

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# Or “how maths can increase the dramatic tension of Reality TV competitions”.

SUGGESTED LEVEL: UPPER SECONDARY

Below is a hierarchy of increasingly dramatic (and mathematically complicated) systems for a reality TV competition such as “Britain’s Got Talent” or “Strictly Come Dancing” to announce their competition winner. Just one rule is needed for creating each system of announcements from the previous one: in mathematics this concept is referred to as **RECURSION**.

**RECURSIVE STEP: to create the next level, we replace each single announcement of the previous level with a set of announcements of all the other contestants!**

Confused? Let’s see how this would work in practice:

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# WHY DO HEXAGONS OCCUR NATURALLY IN NATURE?

This is **Giants Causeway** on the North Coast of Northern Ireland, famed around the world for its awe-inspiring **hexagonal **stone plinths. Incredibly, the stones are built not by a genius mathematician or engineer but by **mother nature**. In this article I’ll explain how it is that nature can afford us such a beautiful display.

The plinths here are in fact igneous **basalt columns**, created when molten lava comes up from inside the earth and cools. As the rock cools it contracts, and this changing shape means that as it solidifies the rock must crack to release the pressure (similar to the way that an ice cube warms, contracts and cracks when you put it into your drink).

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__WHAT’S INTERESTING ABOUT THE NUMBER $2$?__

**TWO **is the only even prime number, and the number of Ronnies in the famous duo. It’s the sides on a digon (two-sided shape such as the panels of a juggling ball), and it’s what it takes to tango. It’s a dual, a duel, a duo and a twin. Two has two **homophones**: ‘to’ and another one ‘too’.

Here are a **couple** of **pairs** of **deuces**:

**CONTINUED FRACTIONS:** consider this fraction:

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