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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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If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

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Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

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Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

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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 challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

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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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Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.

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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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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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There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

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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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This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.

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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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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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What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

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

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

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

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

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

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

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This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

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

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Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

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Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.

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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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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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Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

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I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

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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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Arrange 3 red, 3 blue and 3 yellow counters into a three-by-three square grid, so that there is only one of each colour in every row and every column