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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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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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In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

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Seeing Squares game for an adult and child. Can you come up with a way of always winning this game?

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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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Every day at noon a boat leaves Le Havre for New York while another boat leaves New York for Le Havre. The ocean crossing takes seven days. How many boats will each boat cross during their journey?

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How many different triangles can you make on a circular pegboard that has nine pegs?

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

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Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

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What is the best way to shunt these carriages so that each train can continue its journey?

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How will you go about finding all the jigsaw pieces that have one peg and one hole?

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How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

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This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

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Design an arrangement of display boards in the school hall which fits the requirements of different people.

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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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Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?

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What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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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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Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

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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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Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

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

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Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?

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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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Can you fit the tangram pieces into the outlines of the convex shapes?

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A game for 2 people. Take turns joining two dots, until your opponent is unable to move.

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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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Can you find a way of counting the spheres in these arrangements?

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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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Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

This is the first article in a series which aim to provide some insight into the way spatial thinking develops in children, and draw on a range of reported research. The focus of this article is the. . . .

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How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

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

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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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Billy's class had a robot called Fred who could draw with chalk held underneath him. What shapes did the pupils make Fred draw?

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

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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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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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The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.

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Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?

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An extension of noughts and crosses in which the grid is enlarged and the length of the winning line can to altered to 3, 4 or 5.

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Can you arrange the shapes in a chain so that each one shares a face (or faces) that are the same shape as the one that follows it?

This article looks at levels of geometric thinking and the types of activities required to develop this thinking.

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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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Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.