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Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

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Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . .

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Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

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Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . .

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A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

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Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

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We have exactly 100 coins. There are five different values of coins. We have decided to buy a piece of computer software for 39.75. We have the correct money, not a penny more, not a penny less! Can. . . .

These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.

Here are some examples of 'cons', and see if you can figure out where the trick is.

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Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

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Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

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If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

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There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!

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If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

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Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

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Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

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Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

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I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

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I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

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Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?

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Whenever two chameleons of different colours meet they change colour to the third colour. Describe the shortest sequence of meetings in which all the chameleons change to green if you start with 12. . . .

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Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

This is the second article on right-angled triangles whose edge lengths are whole numbers.

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In the following sum the letters A, B, C, D, E and F stand for six distinct digits. Find all the ways of replacing the letters with digits so that the arithmetic is correct.

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Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

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How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.

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Use your knowledge of place value to try to win this game. How will you maximise your score?

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Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.

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Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.

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Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?

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Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

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Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

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Are these statements relating to odd and even numbers always true, sometimes true or never true?

This article for primary teachers suggests ways in which we can help learners move from being novice reasoners to expert reasoners.

In this article for primary teachers we consider in depth when we might reason which helps us understand what reasoning 'looks like'.

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I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

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In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

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Baker, Cooper, Jones and Smith are four people whose occupations are teacher, welder, mechanic and programmer, but not necessarily in that order. What is each person’s occupation?

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Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?