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

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

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

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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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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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Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .

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Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

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

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?

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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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Imagine you are suspending a cube from one vertex and allowing it to hang freely. What shape does the surface of the water make around the cube?

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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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To avoid losing think of another very well known game where the patterns of play are similar.

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Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.

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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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The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

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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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Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

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Can you find a rule which connects consecutive triangular numbers?

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

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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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I found these clocks in the Arts Centre at the University of Warwick intriguing - do they really need four clocks and what times would be ambiguous with only two or three of them?

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Can you mark 4 points on a flat surface so that there are only two different distances between them?

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

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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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This is a simple version of an ancient game played all over the world. It is also called Mancala. What tactics will increase your chances of winning?

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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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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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The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

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Place a red counter in the top left corner of a 4x4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves. . . .

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Can you make a tetrahedron whose faces all have the same perimeter?

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

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There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you. . . .

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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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P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

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What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

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A cylindrical helix is just a spiral on a cylinder, like an ordinary spring or the thread on a bolt. If I turn a left-handed helix over (top to bottom) does it become a right handed helix?

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A useful visualising exercise which offers opportunities for discussion and generalising, and which could be used for thinking about the formulae needed for generating the results on a spreadsheet.

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Can you mentally fit the 7 SOMA pieces together to make a cube? Can you do it in more than one way?

This article is based on some of the ideas that emerged during the production of a book which takes visualising as its focus. We began to identify problems which helped us to take a structured view. . . .

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Find all the ways to cut out a 'net' of six squares that can be folded into a cube.

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A bicycle passes along a path and leaves some tracks. Is it possible to say which track was made by the front wheel and which by the back wheel?