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

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

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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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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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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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Can you draw a square in which the perimeter is numerically equal to the area?

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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?

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

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

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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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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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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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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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How many models can you find which obey these rules?

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

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Remember that you want someone following behind you to see where you went. Can yo work out how these patterns were created and recreate them?

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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?

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

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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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How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?

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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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These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

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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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How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

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

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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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The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

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Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

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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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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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Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?

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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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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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When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

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

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On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?

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Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

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Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

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Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

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Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

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