I have been writing these blogs for IB Mathematics students for a year and a half now: this is my 36th. They divide into three types: a) General interest and recreation. These blogs look at how mathematics impinges on everyday life and aim to give you a wider appreciation of its relevance. Examples are the probability behind backgammon; Olympic records; some thoughts about zero; a series of blogs about great mathematicians; mathematical puzzles. b) Help with specific areas of mathematics, ...

## A democratic paradox

Democracy depends on people voting for their leaders, but there are many different voting systems in use, all of which aim for a fair result: first past the post; single transferable vote; alternative vote plus are three examples. All have advantages, all have disadvantages. Many popular voting methods use a system which allows voters to select candidates in order of preference. Suppose there are three candidates, A, B and C. You would expect that if more voters preferred A to B, ...

## Coincidence or conspiracy?

Consider these well known historical connections: TITANIC: In 1898, a British author wrote a story about a luxury liner, 800ft long, which was travelling at full speed when it hit an iceberg in the North Atlantic, and sank with the loss of 2500 people. The loss of life was so great because the ship didn't carry enough lifeboats. Its name - The Titan! D-DAY: In May 1944 a schoolteacher, Leonard Dawe, who composed the daily crossword for a London newspaper, included ...

## Practical applications: Vectors

Are vectors just a mathematical idea, or are they of any practical use? Every time you fly anywhere, it's because of vectors that you end up in the right place! Here's why. What is a vector? A vector represents any quantity that has both magnitude and direction. So, vectors can be used to represent displacement (ie change of position), velocity, force - but nor distance, speed or energy. For example, if the distance between two towns along a winding road is 120km, ...

## Some websites to prevent mathematical cobwebs

If you have a long break this (Northern hemisphere) summer, it's important to keep your brain ticking over mathematically so that you don't have to start up from cold when you go back to school next term. I've been on the lookout for some recreational mathematical activity on the web: even just a few minutes each day will be beneficial. I've included a partial screenshot from each site. Pattern recognition Enlarge any one of these 240 patterns. You have to work out ...

## Great Mathematicians 4 – Euclid

Given that Euclid's influence on mathematics, geometry in particular, has never diminished over two thousand years, it is extraordinary that we know so little about his life. He was born around 300BC, and was amongst the first teachers at the great university of Alexandria, founded by Ptolemy I, but it is likely that he studied mathematics in Athens with some of Plato's students. Euclid wrote around a dozen influential mathematical books, but it is his 13 volume treatise The Elements ...

## A chicken pecking problem

The following problem was posed in the 2017 Raytheon MathsCounts national competition: In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to its left or right. What is the expected number of unpecked chicks? The competition is aimed at 13 to 15 year olds, and they have just 45 seconds to answer - the first with the correct solution wins. In this case, a 13 year old Texan boy came up with the answer in ...

## Proving Pythagoras’ Theorem

Hopefully everyone reading this can state Pythagoras' Theorem by heart - but, just in case: "Given a right-angled triangle, the square of the length of the hypotenuse is the sum of the squares of the two shorter sides." Stated more simply, and with the aid of a diagram: c2 = a2 + b2 But how can this result be proved? It's no good drawing lots of right-angled triangles and measuring their sides - firstly, because it's impossible to measure accurately; and secondly, showing that something is true ...

## Understanding significant figures

I'll start with a question - to how many significant figures is the number 500? What does "significant" mean? Consider the number 315 - which digit can you change which would have the least effect on the size of the number? Clearly the 5: increase or decrease it by 1, and 315 changes by 1. But move the 3 up or down 1, and the number will change by 100. So, the 3 is more "significant" than the 5 - and the further ...

## A Human Binary Counter

At the heart of every computer, however complex, are binary numbers - that is, numbers formed only of 0's and 1's. This is because it is easy to represent just two digits electronically: a switch can be on or off; a current can flow one way or the other; a pulse can change a stored digit from one state to the other. How does the binary system work? When we count up to 9 in the decimal system, we've run out ...