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During the third hour after midnight the hands on a clock point in the same direction (so one hand is over the top of the other). At what time, to the nearest second, does this happen?

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Alice's mum needs to go to each child's house just once and then back home again. How many different routes are there? Use the information to find out how long each road is on the route she took.

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60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?

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Stuart's watch loses two minutes every hour. Adam's watch gains one minute every hour. Use the information to work out what time (the real time) they arrived at the airport.

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My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

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On a digital clock showing 24 hour time, over a whole day, how many times does a 5 appear? Is it the same number for a 12 hour clock over a whole day?

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How long does it take to brush your teeth? Can you find the matching length of time?

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Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

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Four numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this sudoku.

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

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

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This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

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The puzzle can be solved with the help of small clue-numbers which are either placed on the border lines between selected pairs of neighbouring squares of the grid or placed after slash marks on. . . .

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Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar".

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Have a go at this game which has been inspired by the Big Internet Math-Off 2019. Can you gain more columns of lily pads than your opponent?

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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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If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

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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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How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

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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?

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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Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

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Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

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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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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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Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.

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Can you use the information to find out which cards I have used?

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A few extra challenges set by some young NRICH members.

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Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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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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How many different symmetrical shapes can you make by shading triangles or squares?

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How will you go about finding all the jigsaw pieces that have one peg and one hole?

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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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Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

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What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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Can you find all the different triangles on these peg boards, and find their angles?

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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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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?

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What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?