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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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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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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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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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How many different triangles can you make on a circular pegboard that has nine pegs?

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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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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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Add or subtract the two numbers on the spinners and try to complete a row of three. Are there some numbers that are good to aim for?

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

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

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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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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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In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

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The pages of my calendar have got mixed up. Can you sort them out?

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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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Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

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What is the best way to shunt these carriages so that each train can continue its journey?

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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?

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

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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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What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

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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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Jack has nine tiles. He put them together to make a square so that two tiles of the same colour were not beside each other. Can you find another way to do it?

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

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What is the date in February 2002 where the 8 digits are palindromic if the date is written in the British way?

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Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

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Design an arrangement of display boards in the school hall which fits the requirements of different people.

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This task encourages you to investigate the number of edging pieces and panes in different sized windows.

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Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

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There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

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Different combinations of the weights available allow you to make different totals. Which totals can you make?

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Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

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Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

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

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.