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

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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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When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

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The Zargoes use almost the same alphabet as English. What does this birthday message say?

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What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?

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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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Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?

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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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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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Six friends sat around a circular table. Can you work out from the information who sat where and what their profession were?

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

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These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.

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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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Lorenzie was packing his bag for a school trip. He packed four shirts and three pairs of pants. "I will be able to have a different outfit each day", he said. How many days will Lorenzie be away?

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This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

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In how many ways can you stack these rods, following the rules?

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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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These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

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My cousin was 24 years old on Friday April 5th in 1974. On what day of the week was she born?

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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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A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

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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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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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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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An investigation that gives you the opportunity to make and justify predictions.

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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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This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?

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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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Find all the numbers that can be made by adding the dots on two dice.

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

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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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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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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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You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

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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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Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

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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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How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

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These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

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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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These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.

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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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Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.