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

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

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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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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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A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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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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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Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

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Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

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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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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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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In this game you are challenged to gain more columns of lily pads than your opponent.

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Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

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An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

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Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold 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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This task encourages you to investigate the number of edging pieces and panes in different sized windows.

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Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

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

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Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?

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

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

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Each of the main diagonals of this sudoku must contain the numbers 1 to 9 and each rectangle width the numbers 1 to 4.

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How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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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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Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.

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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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How many models can you find which obey these rules?

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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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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 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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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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What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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