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Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

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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?

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

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

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Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

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There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?

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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?

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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Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

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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.

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

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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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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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Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

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

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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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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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Can you find the areas of the trapezia in this sequence?

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

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

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Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

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Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?

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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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An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

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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. . . .

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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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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.

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.

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Prove Pythagoras' Theorem using enlargements and scale factors.

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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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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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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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Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

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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?

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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.

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

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Can you explain why a sequence of operations always gives you perfect squares?

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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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I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

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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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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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This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

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

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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?

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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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.