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

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

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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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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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Use the clues about the symmetrical properties of these letters to place them on the grid.

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

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

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

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How many models can you find which obey these rules?

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

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

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When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

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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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Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

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

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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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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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Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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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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Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

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If we had 16 light bars which digital numbers could we make? How will you know you've found them all?