__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.

Read the rest of this page »

__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:

Read the rest of this page »

# 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?)

Read the rest of this page »

__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?

Read the rest of this page »

# 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:

Read the rest of this page »

# 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).

Read the rest of this page »

__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:

Read the rest of this page »

### WHAT’S INTERESTING ABOUT THE NUMBER ONE?

**ONE **is the only number that’s the same in binary, base 10 and Roman Numerals.

It’s the wheels on a **uni**cycle, the rails on a **mono**rail and the players when you go **solo**.

One is the first **odd **number, the first **triangular, square, pentagonal **and **hexagonal **number, and the first **tetrahedral**, **cube** and **Fibonacci **number.

Read the rest of this page »

__WHAT’S THE POINT OF LOGARITHMS?__

On meeting this adorable litter of ten Dalmation Puppies the other day, I quickly spotted (ahem) the need for **Dogarithms **– the canine equivalent of logarithms. The conversation with dog breeder Maxine went something like this:

Read the rest of this page »

**FATHER: “How did you do on your maths test?”**

**JANE: “I did better than average, but then so did most of the class”**

Jane’s statement seem ridiculous – how can *most* people be better than average? Using some mathematical trickery, here’s how she could be logically correct.

Read the rest of this page »