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

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See if you can anticipate successive 'generations' of the two animals shown here.

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Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

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A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?

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A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't. . . .

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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 right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?

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Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

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ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

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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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How can you make an angle of 60 degrees by folding a sheet of paper twice?

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The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?

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Consider a watch face which has identical hands and identical marks for the hours. It is opposite to a mirror. When is the time as read direct and in the mirror exactly the same between 6 and 7?

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Given a 2 by 2 by 2 skeletal cube with one route `down' the cube. How many routes are there from A to B?

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Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

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The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .

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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?

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

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The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

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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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A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?

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Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?

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

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Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

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

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A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.

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Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

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A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?

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

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An irregular tetrahedron has two opposite sides the same length a and the line joining their midpoints is perpendicular to these two edges and is of length b. What is the volume of the tetrahedron?

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A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.

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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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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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What is the shape of wrapping paper that you would need to completely wrap this model?

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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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A huge wheel is rolling past your window. What do you see?

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Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

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Draw all the possible distinct triangles on a 4 x 4 dotty grid. Convince me that you have all possible triangles.

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The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

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A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

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What is the minimum number of squares a 13 by 13 square can be dissected into?

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Can you see how this picture illustrates the formula for the sum of the first six cube numbers?