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

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How many models can you find which obey these rules?

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

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

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

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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 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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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?

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

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

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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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Tim's class collected data about all their pets. Can you put the animal names under each column in the block graph using the information?

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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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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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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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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?