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

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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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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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If these elves wear a different outfit every day for as many days as possible, how many days can their fun last?

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

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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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There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

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

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In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?

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

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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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Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?

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On my calculator I divided one whole number by another whole number and got the answer 3.125. If the numbers are both under 50, what are they?

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Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

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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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Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

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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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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. . . .

This article for primary teachers suggests ways in which to help children become better at working systematically.

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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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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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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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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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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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Exactly 195 digits have been used to number the pages in a book. How many pages does the book have?

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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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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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What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

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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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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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You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

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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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This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.

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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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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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There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?

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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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Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?

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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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Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

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A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

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Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?

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Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.