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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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The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

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The sums of the squares of three related numbers is also a perfect square - can you explain why?

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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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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Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

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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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You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

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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?

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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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Kyle and his teacher disagree about his test score - who is right?

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

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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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The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

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Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

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

This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

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Four jewellers share their stock. Can you work out the relative values of their gems?

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Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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Is the mean of the squares of two numbers greater than, or less than, the square of their means?

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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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I am exactly n times my daughter's age. In m years I shall be ... How old am I?

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The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .

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Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

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What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

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

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Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

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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Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

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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.

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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Can you produce convincing arguments that a selection of statements about numbers are true?

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Can you make sense of these three proofs of Pythagoras' Theorem?

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What is the largest number of intersection points that a triangle and a quadrilateral can have?

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

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

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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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L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?