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Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

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Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?

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Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.

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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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Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

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The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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Katie had a pack of 20 cards numbered from 1 to 20. She arranged the cards into 6 unequal piles where each pile added to the same total. What was the total and how could this be done?

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Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

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This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

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Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

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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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What is the smallest number of coins needed to make up 12 dollars and 83 cents?

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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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Can you rearrange the biscuits on the plates so that the three biscuits on each plate are all different and there is no plate with two biscuits the same as two biscuits on another plate?

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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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Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

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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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Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?

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Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

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Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.

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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?

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

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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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Two sudokus in one. Challenge yourself to make the necessary connections.

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Given the products of diagonally opposite cells - can you complete this Sudoku?

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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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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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Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

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A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

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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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This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find 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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A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly Â£100 if the prices are Â£10 for adults, 50p for pensioners and 10p for children.

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

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

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

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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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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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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 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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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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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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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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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?