Recent Posts by Ian Lucas

Pythagorean Triples

We are all familiar with Pythagoras' Theorem: that if a triangle with sides of length a, b, and c is right-angled, then a2 + b2 = c2. (You can find a proof of the theorem in an earlier blog here). If the sides all have integer values, then the numbers a, b, c form a "Pythagorean triple" - the simplest of which is 3, 4, 5 since 32 + 42 = 52. Further triples can be formed by simple multiplication: thus, 6, 8, 10 ...

An Easy Mathematical Trick

Start by writing the number 1089 on a piece of paper and put it in your pocket. Now get someone to choose a three digit number where the last digit is at least 2 less than the first digit. Turn it round to form a new three digit number, then subtract it from the first one. For example, 481 − 184 = 297. Now turn the new number round to form a fourth three digit number, and add it to the third. 297 + ...

Hypothesis testing made easy

Carrying out a hypothesis test often causes confusion. Here's how it works. Some hypothesis tests start with a known fact, such as "25% of patients treated for a particular disease will suffer side effects." A drug company may then claim that "a new treatment reduces the number of patients suffering side effects." The original figure, the status quo, is known as the "null hypothesis" and given the symbol H0. The new claim is called the "alternative hypothesis" and given the symbol ...

Does division by zero = infinity?

Why should it? Well, try this: 5 ÷ 10 = 0.55 ÷ 1 = 55 ÷ 0.1 = 505 ÷ 0.01 = 5005 ÷ 0.001 = 5000 As we divide by smaller and smaller numbers the result gets ever bigger. Logically, then, as the divisor tends to (ie gets closer to) zero, so the result tends to infinity. But this is not the same as saying that division by zero actually is infinity, is it? What about drawing a graph with ...

How to revise for your Maths exam…

Last month I made some suggestions for a revision plan. Now, having decided you're going to spend a 3 hour session revising Maths, how should you set about it? Since the only thing you will be asked to do in the exam is answer Maths questions, then surely the best way to revise is ... do lots of past paper questions! And you need to have the answers, or the mark scheme, to hand. But it's what you learn from answering ...

Taking your IB Diploma? Here’s a revision plan…

I don't want to be too prescriptive about how you should revise – everyone's different, everyone remembers stuff in different ways – but you should definitely prepare a revision plan. If you don't, then it will be a matter of luck whether or not you'll be ready in time for the exams. Most of you will have at least some holiday time before the exams, and it's imperative you make the most of it – you're on your own, and it's ...

A Mathematical Card Trick

Card tricks fall into several categories, and my favourite are those which look amazing but actually have some pretty simple maths behind them. Here's one of the simplest! 1.     Take an ordinary pack of cards and shuffle it. Deal out 26 cards, face up, and remember the 7th card. 2.    When you've dealt out the 26 cards, pick up the pile and turn them face down on the table, to one side. (If you're brave, shuffle them first, but make ...

Ancient Babylonians Do It Again!

Babylon – in ancient Mesopotamia, now Iraq – hosted one of the earliest recorded civilisations. Partly because they became a trading nation, they developed some of the earliest mathematical techniques. We already know that they were aware of what we now call Pythagoras' Theorem (in a numeric sense, since algebra didn't exist), but the some of the secrets of a tablet known as Plimpton 322 have been unravelled to show that they had developed a highly sophisticated form of trigonometry, ...

Sampling Methods

In statistics, a population is the complete set of data which is to be analysed. A population may consist of people (e.g. those living in a particular city), or living things (e.g. the population of all humpback whales), but could be any set of objects with something in common (e.g. all cars travelling on a particular road in a 24 hour period). Usually, it isn't possible to analyse a complete population. Why? It would take too long It would be too ...

Sam Lloyd, Master of Puzzles

Sam Lloyd was born in Philadelphia in 1841, but lived most of his life in New York. He was primarily a chess player and composer of chess problems, but he also delighted in mathematical puzzling. Actually, not just mathematical puzzles, but word puzzles; picture puzzles; tangrams; he composed thousands of them. In fact, it was one of his pictorial puzzles which grabbed my attention at a very early age, and possibly sparked my enjoyment of such puzzles. Have a look ...

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