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Can you work out how these polygon pictures were drawn, and use that to figure out their angles?

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Join pentagons together edge to edge. Will they form a ring?

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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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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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Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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How much of the square is coloured blue? How will the pattern continue?

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The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

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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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What can you see? What do you notice? What questions can you ask?

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What's the largest volume of box you can make from a square of paper?

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On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

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A visualisation problem in which you search for vectors which sum to zero from a jumble of arrows. Will your eyes be quicker than algebra?

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Can you describe this route to infinity? Where will the arrows take you next?

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A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

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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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A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

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Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

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Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.

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If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?

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Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

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If you move the tiles around, can you make squares with different coloured edges?

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Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

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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 bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?

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Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

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Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

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Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

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A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?

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A train leaves on time. After it has gone 8 miles (at 33mph) the driver looks at his watch and sees that the hour hand is exactly over the minute hand. When did the train leave the station?

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How many different symmetrical shapes can you make by shading triangles or squares?

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In how many ways can you fit all three pieces together to make shapes with line symmetry?

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

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We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

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

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Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

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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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In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

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Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

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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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Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

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

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