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.

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

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

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Prove Pythagoras' Theorem using enlargements and scale factors.

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

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

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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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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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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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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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The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

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

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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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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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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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A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?

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

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

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

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

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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. . . .

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

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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. . . .

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

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

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

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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Prove that the shaded area of the semicircle is equal to the area of the inner circle.

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

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

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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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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. . . .