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ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

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A huge wheel is rolling past your window. What do you see?

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

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The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?

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Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.

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It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

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Prove Pythagoras' Theorem using enlargements and scale factors.

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This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.

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

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Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

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Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

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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. . . .

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

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Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

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It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

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A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?

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

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

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Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

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

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What fractions can you divide the diagonal of a square into by simple folding?

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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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Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

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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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Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

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.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

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Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

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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Can you rearrange the cards to make a series of correct mathematical statements?

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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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Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

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

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Can you make sense of the three methods to work out the area of the kite in the square?

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

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

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

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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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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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The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .

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In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

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.