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Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

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The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

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How many trapeziums, of various sizes, are hidden in this picture?

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Can you see who the gold medal winner is? What about the silver medal winner and the bronze medal winner?

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An activity making various patterns with 2 x 1 rectangular tiles.

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Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

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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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These practical challenges are all about making a 'tray' and covering it with paper.

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Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

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Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

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In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

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Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

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These rectangles have been torn. How many squares did each one have inside it before it was ripped?

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How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

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What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

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If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

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How many different triangles can you draw on the dotty grid which each have one dot in the middle?

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This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

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Can you fill in the empty boxes in the grid with the right shape and colour?

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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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How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?

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

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Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

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Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

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Investigate the different ways you could split up these rooms so that you have double the number.

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Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

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Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

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Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

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Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

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A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

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

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My coat has three buttons. How many ways can you find to do up all the buttons?

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This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

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Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

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My briefcase has a three-number combination lock, but I have forgotten the combination. I remember that there's a 3, a 5 and an 8. How many possible combinations are there to try?

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A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

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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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Can you find all the different triangles on these peg boards, and find their angles?

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How many models can you find which obey these rules?

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Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

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Can you use the information to find out which cards I have used?

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How can you put five cereal packets together to make different shapes if you must put them face-to-face?

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Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

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What could the half time scores have been in these Olympic hockey matches?