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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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A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

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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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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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Imagine a stack of numbered cards with one on top. Discard the top, put the next card to the bottom and repeat continuously. Can you predict the last card?

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The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

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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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Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

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

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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?

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

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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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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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On a clock the three hands - the second, minute and hour hands - are on the same axis. How often in a 24 hour day will the second hand be parallel to either of the two other hands?

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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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Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

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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 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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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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Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.

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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 cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

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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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If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

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

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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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How many moves does it take to swap over some red and blue frogs? Do you have a method?

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Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

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

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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?

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

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