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

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

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A cube is made from smaller cubes, 5 by 5 by 5, then some of those cubes are removed. Can you make the specified shapes, and what is the most and least number of cubes required ?

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

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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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ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.

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

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

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

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

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

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

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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 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 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?

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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?

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

Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.

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

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

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Some treasure has been hidden in a three-dimensional grid! Can you work out a strategy to find it as efficiently as possible?

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

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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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This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at.

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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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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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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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Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube so that the surface area of the remaining solid is the same as the surface area of the original?

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

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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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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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These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?