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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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Find out about the lottery that is played in a far-away land. What is the chance of winning?

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Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

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Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

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Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

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

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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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If you had 36 cubes, what different cuboids could you make?

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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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Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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

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How many models can you find which obey these rules?

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

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

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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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This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

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Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

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10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

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

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

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How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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

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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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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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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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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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Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

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Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

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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 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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How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

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In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?

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Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

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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Pentagram Pylons - can you elegantly recreate them? Or, the European flag in LOGO - what poses the greater problem?

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In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

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Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.

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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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This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

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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.

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George and Jim want to buy a chocolate bar. George needs 2p more and Jim need 50p more to buy it. How much is the chocolate bar?