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# David(David3142)

Joined 9 years ago from West Midlands, England Last activity 7 days ago

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• 0

### What Is the Sum of the Sequence 1, 1/2, 1/4, 1/8, 1/16, ...?

4 days ago

Take the sequence of numbers 1, 1/2, 1/4, 1/8, 1/16, etc where each number is half of the preceding number. What happens if we try to add these together? What is the total sum? Find out in this quick maths article.

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### Is There a Biggest Prime Number or Do They Continue Infinitely?

6 weeks ago

Is there a largest prime number, or are there infinitely many primes? We're going to take a look at what prime numbers are and whether a largest one exists or not.

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### How to Measure the Height of a Tall Building or Mountain

2 months ago

Find out how to measure the height of a tall structure, mountain or any other high object without having to climb it with a tape measure. All of this using the power of mathematics.

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### How to Draw Pie Charts

2 months ago

A quick maths lesson covering how to draw pie charts from given data. Find out how to calculate the angle of each slice and how to draw them in a circle to construct a pie chart.

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### Maths Olympiad: Find the Answer to 8×9×10×11×12 Without Using a Calculator

4 months ago

Finding the product of 8, 9, 10, 11 and 12 without the use of calculator may seem like a daunting task, but can be made easier with some some simple maths tricks. Find out how to solve this Maths Olympiad question in this article.

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### Find Four Primes Smaller Than 100 Which Are Factors Of 3^32 − 2^32: Maths Olympiad Walkthrough

4 months ago

We need to find four prime numbers smaller than 100 which are factors of 3^32 − 2^32. Find out how to solve this Maths Olympiad question in this walkthrough article.

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### How to Multiply Decimal Numbers Without a Calculator

5 months ago

Find out how to multiply decimal numbers by whole numbers and by other decimals in this quick maths how-to article.

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### How To Find 3^9 Without Using a Calculator - Number Problems

6 months ago

How can we find the value of 3^9 without using a calculator or even a written method? Find out how in this quick maths article.

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### What Is the Maximum Score in Ten-Pin Bowling?

6 months ago

Have you ever wondered what the maximum score achievable in ten-pin bowling is? Find the answer to this and how to calculate it in this article.

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### How Likely Are You to Hit the Centre of the Archery Target?

6 months ago

Have you ever wondered what the probability of hitting the centre of the archery target with a bow and arrow is? In this article we will look at the dimensions of the target and find out just how likely you are to score a perfect 10.

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### How to Complete the Square With Algebraic Expressions

7 months ago

Find out how to simplify algebraic expressions and equations by completing the square in this quick maths how-to article.

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### Different Kinds of Prime Numbers: Twin, Cousin and Sexy Primes

3 months ago

Most people have come across prime numbers, but have you heard of twin primes, cousin primes and sexy primes? Read on to find out more about these special primes.

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### What Is Pi? A Short History of This Incredible Number

4 months ago

What is pi, how was it calculated and what are some uses for this incredible number? Find out the answers to these and more in this article on the history of pi.

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### Edexcel GCSE Maths Exam 2023 - Solving the Final Octagon Question

4 months ago

The final question on paper 2 for the Edexcel GCSE maths exam 2023 is definitely a tricky one. Here's how to solve it.

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### How To Solve the GCSE Maths Question That's Leaving Parents Stumped

4 months ago

This GCSE maths question has been leaving parents baffled, so we're going to have a look at how to answer it here.

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### How to Find Equivalent Fractions

4 months ago

Finding equivalent fractions is useful when comparing fractions or performing fraction arithmetic. Find out how to convert fractions into equivalent forms in this quick guide.

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### What Is the Unexpected Hanging Paradox?

6 months ago

The unexpected hanging paradox (also known as the surprise test paradox) is a problem involving expectations of the timing of a future event, which lead to a rather surprising paradox. Read on to find out more.

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### How to Add, Subtract, Multiply and Divide Fractions

4 months ago

A guide to performing basic arithmetic with fractions. Learn how to add, subtract, multiply and divide fractions with each other and with whole numbers.

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### What Is the Base Rate Fallacy? Examples of Probability Problems

3 months ago

If a test for a disease comes back positive, what is the actual probability of you being infected? It's often different from what you may think because of a thing called the Base rate fallacy. Find out how this works in this quick maths article.

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### Every Prime Number Larger Than 3 Is 1 Away From a Multiple of 6: A Proof

4 months ago

Did you know that every prime number larger than 3 is only 1 away from a multiple of 6? In this quick maths guide, we are going to prove this surprising fact.

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### How to Integrate by Parts: Calculus Help

2 months ago

Struggling with integration by parts? Not sure how it works? Find out how to integrate by parts and how to spot when you should integrate by parts in this quick guide.

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### Espresso Drinks Menu: A Quick Beginner's Guide

15 months ago

Have you ever been confused by the sheer amount of choice on a coffee shop menu? What is the difference between a flat white and an Americano with milk? Learn this and more in this quick guide.

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### How to Integrate 1/(e^x + e^-x) With Respect to x

4 months ago

Find out how to integrate 1/(e^x + e^-x) with respect to x in this quick how-to maths article.

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### How Do You Write 1999 in Roman Numerals?

4 months ago

Roman numerals are still used a surprising amount and most people are comfortable with the basics, but what about writing a large number like 1999 in numerals? Find out how we do this in this quick article.

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### 5 Shortest Serving British Prime Ministers

4 months ago

Liz Truss resigned as the Prime Minister of the UK after only 55 days in office, but is she the shortest serving of all? This article covers British Prime Ministers that served the shortest time.

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### How to Solve Direct Proportion Questions: Maths Help

18 months ago

A quick guide to solving direct proportion questions including finding missing values in tables and solving worded problems.

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### How to Make a Mathematical Paper Snowflake (Christmas Crafts)

19 months ago

Make a symmetrical Christmas snowflake using just paper, pencil, a pair of compasses, scissors and a straight edge with these quick instructions. Perfect for homemade Christmas decorations!

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### What Is the Collatz Conjecture? Unsolved Mathematical Problems

3 months ago

Find out all about the Collatz Conjecture, its origins and more in this quick maths article.

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### How to Do Long Multiplication Using Napier's Method

4 months ago

Find out how to do long multiplication using Napier's Method, also known as Lattice multiplication, in this quick maths article.

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### How to Answer the 'Impossible' Question on the Edexcel GCSE Maths Paper 2022

4 months ago

The final question on the 2022 Edexcel GCSE Mathematics paper has been branded as 'impossible' by many people. In this article we take a look at how to solve it.

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### How to Expand a Pair of Brackets, an Algebra Walkthrough

4 months ago

Learn how to expand a pair of brackets such as (x+1)(x+3) in this quick algebra maths lesson.

• 0

### How to Find the Averages and Range From Grouped Data

4 months ago

Learn how to find the mean, median, mode and range from grouped data in a frequency table in this quick maths article.

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### How to Find the Average From a Frequency Table

4 months ago

Learn how to find the mean, median, mode and range from a frequency table in this quick maths article.

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### How to Find the Average of a Group of Numbers

4 months ago

Find out how to calculate the different averages and the range of a group of numbers in this quick maths article all about the mean, median, mode and range.

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### How Many Gifts Did I Get Over the Twelve Days of Christmas?

4 months ago

Find out just how many gifts you would receive if your true love actually gave you all of the gifts from the Twelve Days of Christmas carol.

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### How to Use Pythagoras' Theorem to Find Missing Sides on Right-Angled Triangles

4 months ago

Find out how to use Pythagoras' theorem to find the length of missing sides on right-angles triangles in this quick maths article.

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### How to Find the Lowest Common Multiple and Highest Common Factor of Two Numbers

4 months ago

In this quick maths article we look at how to find the lowest common multiple and highest common factor of two numbers quickly and efficiently using prime factors.

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### How to Write a Number as a Product of Its Prime Factors

8 weeks ago

In this quick maths article, find out what prime factors are, how to find them using a factor tree and how any number can be written as a product of prime factors.

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### What Do the Angles in a Polygon Add Up To?

4 months ago

In this quick maths article, find out what the angles in a polygon add up to, as well as the difference between interior and exterior angles.

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### What Are Triangular Numbers?

4 months ago

What are the triangular numbers? Find out how the sequence is defined and more interesting facts in this quick maths article.

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### Why Do We Split a Circle Into 360 Degrees? A Look at the Origins of 360

4 months ago

Ever wondered why circles have 360 degrees? Why not a simpler number such as 10 or 100? Find out some possible reasons in this quick maths article.

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### How to Factorise a Quadratic Algebraic Equation

4 months ago

How do you factorise quadratic equations quickly and easily? Find out in this quick algebra maths lesson.

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### How to Prove Pi Equals 2: A Mathematical Paradox

3 months ago

Do you know what pi equals? 3.141 ... right? Here's a curious geometric proof that pi equals 2.

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### How to Calculate a Percentage of an Amount Using a Decimal Multiplier

4 months ago

Learn how to calculate a percentage of an amount quickly and easily by using the decimal multiplier method.

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### What Is a Dudeney Number?

4 months ago

Learn about Dudeney numbers and how to find them in this quick maths guide.

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### How to Solve a Quadratic Equation: 3 Methods

4 months ago

Learn three ways of solving quadratic equations in this quick maths article, including factorising the equation, completing the square and the quadratic formula.

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### How to Find the Sum of a Geometric Sequence

3 months ago

How do we find the sum of a geometric sequence? Discover how to quickly add up the first terms of a geometric sequence in this quick maths article.

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4 months ago

What are hexadecimal numbers, what are they used for and how do we convert them into decimal numbers? Find out all of this and more in this quick math article.

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### How Many Socks Make a Pair? A Mathematics Problem

4 months ago

You pull socks out of a drawer, desperately trying to find a matching pair, but all you get are odd socks. How many socks will you need to pull out to be guaranteed a matching pair?

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### A Quick Way to Solve 1000^2 − 999^2: The Difference of Two Squares

3 months ago

A look at how using the difference of two squares can speed up your mental arithmetic with large numbers.

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### Zeno's Paradox: Achilles and the Tortoise

6 months ago

Zeno's paradox tells the story of a race between the Greek hero Achilles and a tortoise but with a surprising twist. Find out more in this quick article.

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### How to Prove That the Square Root of 2 Is Irrational

3 months ago

It's common knowledge that the square root of two is an irrational number, but how can we prove it? Find out in this quick mathematics article.

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### Rationalizing the Denominator: Dealing With Surds/Radicals in Fractions

4 months ago

Struggling with surds and radicals in fractions? You're not alone. I'll explain it to you. Learn how to rationalize the denominator in this quick high school mathematics article.

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### Why Do We Rationalise the Denominator? Standard Notation While Using Surds/ Radicals

4 months ago

When learning about surds/radicals at school, we are always told to rationalise the denominator. But what does this mean and why do we do it? Find out why in this quick mathematics article.

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### The Prisoner's Dilemma: Using Game Theory to Achieve the Optimal Solution

4 months ago

Two criminals have been arrested for a crime, but the police don't have enough evidence to convict. They offer each criminal the chance to reduce their sentence by testifying against the other, but is this the best option? Find out in this quick math article about the prisoner's dilemma.

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### Decreasing the Circumference of Differently Sized Circles: A Counterintuitive Cricket Problem

3 years ago

The village of Smallville and the Intergalactic Gods have both had five metres of boundary rope stolen from their circular cricket pitches. They have vastly different sized pitches, so whose pitch will reduce in radius the most?

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### Being Careful When You Average an Average: A Basketball Problem

22 months ago

Alan has the best scoring average in the first half of the basketball season and the best scoring average for the second half of the season, but when everything is tallied together he finds that it is Brian who has the highest average over the season. Find out how in this quick math article.

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### Hilbert's Paradox of the Grand Hotel: Another Look at Infinity

4 months ago

A look at Hilbert's Paradox of the Grand Hotel (also known as the 'Infinite Hotel Paradox') which uses hotel rooms to give a new twist to infinite numbers.

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### How Large Is Infinity?

4 months ago

What do you think of when you hear the word infinity? Can there be different sizes of infinity? Find out in this quick math article.

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### Why Do We Have Leap Years?

6 months ago

Ever wondered why we have leap years? Why do we add an extra day to February every four years? Find out why and learn about the history of calendars and leap years in this article.

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### How to Add the Numbers 1-100 Quickly: Summing Arithmetic Sequences

4 months ago

If you were asked to add together all of the numbers from 1 to 100, what would you do? Would you go through each number, adding to your total as you went or is there a quicker way? Find out in this article.

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### N-bonacci Sequences - Taking Fibonacci Further

3 years ago

Most of us have heard of Fibonacci numbers and know a little about what they are, but what about N-bonacci numbers? In this article we will look at what they are and how they link to the Fibonacci sequence.

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### Mathematical Numbers: What Is 'e'?

4 months ago

The number 'e' is an extremely important number in mathematics and is used in various situations. But what is it? How do we calculate it? Find out in this article.

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### The Maths Behind Dating: Optimal Stopping Theory

10 months ago

When dating, should you choose to settle down with the first person you date, or should you get a few dates under your belt before choosing? Find out what the maths says in this fun article about the optimal stopping theory.

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### Five More of History's Most Influential Women in STEM

3 years ago

A follow up to my original article on influential women in STEM, featuring another five incredible minds whose contribution to science, technology, engineering and maths changed our world forever.

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### Five of History's Most Influential Women in STEM

3 years ago

Histoy is full of many inspiring individuals who have worked in the fields of Science, Technology, Engineering and Mathematics (STEM). In this article we take a look at five of the most influential women from this list.

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### Why You Should Always Order the Large Pizza

3 months ago

Ever wondered if you should order one large pizza or two medium pizzas? Let some simple mathematics find the answer.

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### Why Does Time Slow Down as You Approach the Speed of Light?

4 months ago

Why does time slow down as you approach the speed of light? Find out why in this article featuring Einstein, space travel, light clocks and more.

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### Three of Isaac Newton's Most Important Contributions to the World

3 years ago

Isaac Newton is one of the most influential and famous mathematicians and scientists who has ever lived. In this article read about some of his most well-known and important contributions to the world.

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### The Maths Behind A4 Paper

4 months ago

Look at how A4 paper is defined and what makes it so interesting mathematically.

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### Interesting Facts About Pascal's Triangle

4 months ago

What is Pascal's Triangle, and why is it so interesting? Find out here.

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### What Is an Erdős Number?

4 months ago

Find out about Paul Erdős, his life and the number that bears his name.

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### 3 Interesting Fractals From Koch, Sierpinski and Cantor

4 months ago

Here is a look at what fractals are and three of the most popular examples: the middle third Cantor Set, the Koch curve, and the Sierpinski triangle.

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### How Do Binary Numbers Work?

4 months ago

Learn how the binary system works, how to convert between binary and decimal and why the binary system is so important.

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### Four Interesting Types of Mathematical Numbers

4 months ago

There are many unique numbers beyond the ones we commonly learn about. The more rarely discussed types include vampire numbers, narcissistic numbers and more.

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### The Handshake Problem: Creating a Mathematical Solution

4 months ago

If you have a room full of people, how many handshakes are required before everybody has shaken hands with everybody else exactly once?

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### 'The Andromeda Evolution' by Daniel H. Wilson: A Book Review

6 months ago

This article contains a book review of 'The Andromeda Evolution' by Daniel H.Wilson, a follow-up to the bestselling 'The Andromeda Strain' by Michael Crichton.

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### Post-Workout Cacao and Maca Smoothie Recipe

23 months ago

This article contains a recipe for my favourite post-workout smoothie containing cacao, maca and banana.

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### Speed Arithmetic - How to Multiply by 11 Without a Calculator

4 years ago

How to use a quick and easy method to multiply by 11 without using a calculator.

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### Speed Arithmetic - How to Multiply and Divide by 5 Without a Calculator

4 years ago

How to use a simple mental arithmetic method for quickly multiplying and dividing by 5 in your head.

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### How to Differentiate From First Principles

4 months ago

Differentiation is used to find the rate of change of a mathematical function as its input changes. Read on to find out how to differentiate from first principles.

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### How to Calculate Compound Percentage Changes

20 months ago

How to calculate compound interest increases and decreases using the multiplier method.

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### How Many Squares Are on a Chessboard? A Maths Problem

4 months ago

How many squares of all different sizes are there on a chessboard? Instead of counting them up, we can calculate the answer with a formula.

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### Bertrand's Paradox: A Problem in Probability Theory

4 months ago

A look at Bertrand's Paradox, where a seemingly simple probability question yields some very different results.

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### How Do Scale Factors Work for Area and Volume?

4 months ago

Wondering how to use scale factors when working with area and volume? Read on for a full explanation of scale factor formulas, plus examples.

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### Pythagoras' Theorem - A Proof

2 years ago

How to prove Pythagoras' Theorem using geometry and some simple algebra.

• 0

### How to Find Pi Using Regular Polygons

4 months ago

An article about how to find the value of Pi geometrically using regular polygons.

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### Maximizing the Area of a Rectangle

4 months ago

A farmer has 100 meters of fence and wants to make a rectangular enclosure with the largest possible area. What size should the sides of his rectangle be? How do you maximize the area of the rectangle?

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4 months ago

How many people do you need to have in a room before you have a better than 50% chance of at least two people sharing a birthday? The answer is surprisingly low.

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### The Monty Hall Problem: a Probability Problem With Goats and Cars

2 months ago

The Monty Hall Problem (also known as the probability problem with the goats) is a popular example of how intuition can often lead you to make mistakes in probability.

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### Why Do Buses Come in Threes?

3 years ago

You sit around for ages waiting for a bus and then three come along at once. But why? Find out why buses come in threes in this quick maths article.

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### How to Calculate the Probability of Winning the National Lottery in the UK

4 months ago

This is an article about the probability of winning the National Lottery jackpot or its smaller prizes.

• 3

### Rice on a Chessboard - Exponential Numbers

2 months ago

A story about the surprising power of exponential numbers.

• 0

### How to Write Fractions in Microsoft Word

14 months ago

For basic fractions, go to the 'Insert' tab and click 'Symbol'. By scrolling down, you will be able to find several simple fractions ready for you to insert.

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### How to Hide Error Values in Microsoft Excel

14 months ago

This is a tutorial on how to hide error values such as N/A in Microsoft Excel.

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### How to Use the VLookup Function in Microsoft Excel

14 months ago

This is an instructional article on how to use the VLookup Function in Microsoft Excel

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### How to Change a Microsoft PowerPoint Slideshow Into a Video

12 months ago

Step-by-step instructions for converting a Microsoft PowerPoint slideshow into video, including adding sound.

• 3

### Honest Book Review: "Mr. Mercedes" by Stephen King

6 months ago

Love Stephen King? See my review of "Mr. Mercedes," a hard-boiled detective novel that might just be your next page-turner.

• 1

### Book Review: Kerrigan in Copenhagen by Thomas E. Kennedy

5 years ago

Kerrigan in Copenhagen by Thomas E. Kennedy is a guide book to Copenhagen, especially it's drinking establishments, disguised as a novel, following Kerrigan as he meanders from bar to bar, discussing literature, life and jazz.

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### Teaching Speed-Time Graphs

5 years ago

An outstanding lesson plan for introducing speed-time graphs to your students.

• 1

3 years ago

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### The Real Ghostbusters

5 years ago

The Real Ghostbusters - A classic cartoon from the 80s.

• 0

### The Very Best of Modern Talking - An Album Review

3 years ago

The Very Best of Modern Talking, greatest hits album - A review

• 0

### My Favourite Tower Defence Games

5 years ago

A short article about my favourite web-based tower defence games.

• 0

### How to Cure a Blocked/Stuffy Nose

4 years ago

As a teacher, I regularly catch colds and other bugs. Generally, they're not particularly bad and I can work through them, but sometimes I get hit by a heavy cold and end up with an extremely runny nose. After a day or two this then turns into a...

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### Favourite Cartoons of My Childhood

5 years ago

A list of classic cartoons from my childhood (late 80s, early 90s).

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### My Favourite Mathematics Authors

4 years ago

Popularisers of maths, such as Iain Stewart and Marcus de Sautoy, help to bring the magic and wonder of mathematics to everybody, regardless of their level of maths education.

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### My Favourite Scandinavian Pop Groups

5 years ago

There are a surprisingly large number of hugely popular music acts from Scandinavia. Read on to find out more.

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### My Favourite Console - Commodore Amiga 600

4 years ago

All about my very first console, The Commodore Amiga 600.

• 1

### My Favourite A-Team Memorabilia

4 years ago

A Hubpage all about fantastic A-Team memorabilia. Relive those 80's moments.

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### Tomato and Hotdog Pasta Recipe

5 years ago

This is my favourite quick pasta recipe for when I'm home alone. I'm not much of a cook and I created this at university, so it's simple and easy to do and the ingredients are cheap and commonly found in most kitchens. I often make this if I'm...

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### The Nightmare Before Christmas - Family Halloween Movie

2 years ago

The Nightmare Before Christmas - A review of the perfect family Halloween movie.

• 0

### Shakin' Stevens Greatest Hits

6 years ago

Shakin' Stevens Greatest Hits - An album review

• 2

### Rules of Honour - Matt Hilton

4 years ago

Rules of Honour by Matt Hilton - A review.

• 6

### Flash Gordon - Film review

4 years ago

Film review - Flash Gordon, a cult classic movie from the 80's.

• 0

### My Family and Other Animals - Gerald Durrell

4 years ago

My Family and Other Animals by Gerald Durrell - A book review.

• 0

### Teaching Circumference and Diameter of a Circle - Lesson Ideas

2 months ago

Some lesson ideas for teaching the circumference and diameter of a circle. An excellent introduction to pi.

• 4

### The Traveling Wilburys -Vol. 1

2 years ago

The Traveling Wilburys - Vol. 1, an album review.

• 2

### Hunky Dory - David Bowie

4 years ago

Hunky Dory by David Bowie - A Review.

• 0

### Doctor Who - The Vault

6 years ago

Doctor Who - The Vault, a book review.

• 2

### Classic 80s Tv - the A-Team

4 years ago

The A-Team; my favourite tv program of all time. Relive some classic 80's moments here.

• 3

2 years ago

• 0

### 'Actually', Pet Shop Boys

6 years ago

'Actually' by the Pet Shop Boys - A review

• 3

### The Night Before Christmas Book Gift

5 years ago

An article about the classic book 'The Night Before Christmas'.

• 0

### The Mentalist on DVD

4 years ago

The Mentalist - One of my favourite detective shows, available on DVD.

• 0

### Things to Do in Bruges, Belgium

4 years ago

Top places to visit in Bruges, Belgium. A list of tourist attractions for visitors to this beautiful Belgian city.

• 8

### Jurassic Park Book Review

16 months ago

Jurassic Park by Michael Crichton - A book review.

• 5

3 years ago

A list of movies inspired by Stephen King novels.

• 2

### My Top Five Favourite Stephen King Novels

4 years ago

A list of my favourite five Stephen King Novels

• 4

### A Beginner's Guide to Reality - Book Review

3 years ago

A Beginner's Guide to Reality by Jim Baggot - A review.

• 0

### Using the HTML Element on Weebly

4 years ago

So you've got your Weebly website up and running using the basic elements such as text and image. Now you're wondering how to use the HTML element. There are many things you can do with this useful element and in this lens I will show some of the...

• 6

### Learning the Presidents of the United States

4 years ago

An article all about my attempt to memorise all of the Presidents of the United States, including some interesting trivia along the way.

• 0

### Halloween Films for the Whole Family

5 years ago

Kids love Halloween. It's a great excuse to dress up in a fancy costume and then obviously there's the chocolate and sweets. There are also some great Halloween films out there to get the family in the mood; but what's a good choice for the family?...

• 1

### The Best Tv Cars and Vehicles of the 80s

4 years ago

A list of some of my favourite cars and other vehicles from classic 80's tv series.

• 3

### Cold Relieving Lemon, Garlic and Ginger Drink

3 years ago

How to make the perfect cold relieving drink using lemon, garlic, ginger and honey.

• 2

### Ideas for Maths Lesson Starters

7 months ago

Having difficulties finding interesting starters for your maths lessons? Then check out the ideas in this article that can help get your students' brains ready for maths with these lesson starters.

• 0

### My Favourite Modern Thriller Writers

5 years ago

A list of my favourite modern thriller writers including Lee Child and Matt Hilton.

• 0

### Great Ideas for Hands-On Maths Lessons

5 years ago

A collection of ideas for teaching maths in a fun, hands-on way.

• 3

### Shrove Tuesday or Pancake Day

2 years ago

A quick article all about Shrove Tuesday/Pancake day, its history and how to make your own pancakes

• 7

### My Favourite Things to Do in Salzburg, Austria

3 years ago

A list of great places to visit in Salzburg, Austria; home of Wolfgang Amadeus Mozart.

• 6

### Fried Egg in a Basket Recipe

3 years ago

How to make fried egg in a basket, a delicious snack using eggs and bread.

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### Fishfinger (Fishstick) and Cheese Toastie Sandwich Recipe

4 years ago

How to make a fishfinger and cheese toastie sandwich.

• 2

### Chocolate Biscuit Loaf Recipe

4 years ago

How to make delicious chocolate biscuit loaf, ideal for parties or snacking.

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