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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?

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Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

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

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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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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?

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

Challenge Level

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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A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

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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.

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The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

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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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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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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?

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

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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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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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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?

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Can you find a rule which connects consecutive triangular numbers?

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Can you find a rule which relates triangular numbers to square numbers?

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Show that all pentagonal numbers are one third of a triangular number.

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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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How good are you at finding the formula for a number pattern ?

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Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

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Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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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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How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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A box has faces with areas 3, 12 and 25 square centimetres. What is the volume of the box?

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The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?

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Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

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

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

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

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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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Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

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

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

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

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

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Can you see how to build a harmonic triangle? Can you work out the next two rows?