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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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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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Winifred Wytsh bought a box each of jelly babies, milk jelly bears, yellow jelly bees and jelly belly beans. In how many different ways could she make a jolly jelly feast with 32 legs?

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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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This task follows on from Build it Up and takes the ideas into three dimensions!

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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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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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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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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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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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Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

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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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Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number.

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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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Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

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What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.

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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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This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

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You have 5 darts and your target score is 44. How many different ways could you score 44?

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Surprise your friends with this magic square trick.

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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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Start with four numbers at the corners of a square and put the total of two corners in the middle of that side. Keep going... Can you estimate what the size of the last four numbers will be?

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In this section from a calendar, put a square box around the 1st, 2nd, 8th and 9th. Add all the pairs of numbers. What do you notice about the answers?

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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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If you had any number of ordinary dice, what are the possible ways of making their totals 6? What would the product of the dice be each time?

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Throughout these challenges, the touching faces of any adjacent dice must have the same number. Can you find a way of making the total on the top come to each number from 11 to 18 inclusive?

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

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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?

This article for primary teachers encourages exploration of two fundamental ideas, exchange and 'unitising', which will help children become more fluent when calculating.

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The clockmaker's wife cut up his birthday cake to look like a clock face. Can you work out who received each piece?

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This dice train has been made using specific rules. How many different trains can you make?

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Can you make square numbers by adding two prime numbers together?

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In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

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Ben has five coins in his pocket. How much money might he have?

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Arrange three 1s, three 2s and three 3s in this square so that every row, column and diagonal adds to the same total.

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Put a number at the top of the machine and collect a number at the bottom. What do you get? Which numbers get back to themselves?

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These two group activities use mathematical reasoning - one is numerical, one geometric.

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Annie cut this numbered cake into 3 pieces with 3 cuts so that the numbers on each piece added to the same total. Where were the cuts and what fraction of the whole cake was each piece?

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Find out what a Deca Tree is and then work out how many leaves there will be after the woodcutter has cut off a trunk, a branch, a twig and a leaf.

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This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!

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Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether.

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Can you each work out the number on your card? What do you notice? How could you sort the cards?

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On the table there is a pile of oranges and lemons that weighs exactly one kilogram. Using the information, can you work out how many lemons there are?

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Rocco ran in a 200 m race for his class. Use the information to find out how many runners there were in the race and what Rocco's finishing position was.

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I was looking at the number plate of a car parked outside. Using my special code S208VBJ adds to 65. Can you crack my code and use it to find out what both of these number plates add up to?

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This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

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Well now, what would happen if we lost all the nines in our number system? Have a go at writing the numbers out in this way and have a look at the multiplications table.

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Investigate the totals you get when adding numbers on the diagonal of this pattern in threes.

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Can you score 100 by throwing rings on this board? Is there more than way to do it?