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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.

Challenge Level

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

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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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Describe how to construct three circles which have areas in the ratio 1:2:3.

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Drawing a triangle is not always as easy as you might think!

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You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

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Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

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Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

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Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

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Can you work out the side length of a square that just touches the hypotenuse of a right angled triangle?

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The farmers want to redraw their field boundary but keep the area the same. Can you advise them?

Challenge Level

Draw a line (considered endless in both directions), put a point somewhere on each side of the line. Label these points A and B. Use a geometric construction to locate a point, P, on the line,. . . .

Challenge Level

Construct a line parallel to one side of a triangle so that the triangle is divided into two equal areas.

Challenge Level

Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

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How can you represent the curvature of a cylinder on a flat piece of paper?

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Explore patterns based on a rhombus. How can you enlarge the pattern - or explode it?

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The challenge is to produce elegant solutions. Elegance here implies simplicity. The focus is on rhombi, in particular those formed by jointing two equilateral triangles along an edge.

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Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

Jenny Murray describes the mathematical processes behind making patchwork in this article for students.

Challenge Level

Design and construct a prototype intercooler which will satisfy agreed quality control constraints.

Challenge Level

What shape and size of drinks mat is best for flipping and catching?