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

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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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Can you find a rule which connects consecutive triangular numbers?

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

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

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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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We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

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Make some loops out of regular hexagons. What rules can you discover?

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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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Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

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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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Kyle and his teacher disagree about his test score - who is right?

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

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Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

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

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

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

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There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

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Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

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

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Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

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

This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.

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

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What are the possible dimensions of a rectangular hallway if the number of tiles around the perimeter is exactly half the total number of tiles?

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The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

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

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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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Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

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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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Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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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. . . .