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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.

Challenge Level

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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A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

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If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

Challenge Level

If you know the perimeter of a right angled triangle, what can you say about the area?

Challenge Level

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

Challenge Level

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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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?

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A job needs three men but in fact six people do it. When it is finished they are all paid the same. How much was paid in total, and much does each man get if the money is shared as Fred suggests?

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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. . . .

Challenge Level

Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?

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Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

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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. . . .

Challenge Level

A task which depends on members of the group noticing the needs of others and responding.

Challenge Level

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

Challenge Level

If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?

Challenge Level

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

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Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

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

Challenge Level

Surprising numerical patterns can be explained using algebra and diagrams...

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What is special about the difference between squares of numbers adjacent to multiples of three?

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What is the total number of squares that can be made on a 5 by 5 geoboard?

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Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

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

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Can you explain what is going on in these puzzling number tricks?

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

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Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

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Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Challenge Level

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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Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

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There are unexpected discoveries to be made about square numbers...

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Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

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Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

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Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

Challenge Level

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Challenge Level

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Challenge Level

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

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Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

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The Number Jumbler can always work out your chosen symbol. Can you work out how?

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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.

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Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

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

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A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

Challenge Level

Play around with the Fibonacci sequence and discover some surprising results!

Challenge Level

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .