The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.

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Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

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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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Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

This game for two, was played in ancient Egypt as far back as 1400 BC. The game was taken by the Moors to Spain, where it is mentioned in 13th century manuscripts, and the Spanish name Alquerque. . . .

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This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

An ancient game for two from Egypt. You'll need twelve distinctive 'stones' each to play. You could chalk out the board on the ground - do ask permission first.

A game for two players based on a game from the Somali people of Africa. The first player to pick all the other's 'pumpkins' is the winner.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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A game for 2 people. Take turns joining two dots, until your opponent is unable to move.

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Can you make a tetrahedron whose faces all have the same perimeter?

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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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Some treasure has been hidden in a three-dimensional grid! Can you work out a strategy to find it as efficiently as possible?

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A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

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The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

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Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

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Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

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Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.

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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?

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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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How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?

This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .

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This is a simple version of an ancient game played all over the world. It is also called Mancala. What tactics will increase your chances of winning?

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The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

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Investigate how networks can be used to solve a problem for the 18th Century inhabitants of Konigsberg.

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If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

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Place a red counter in the top left corner of a 4x4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves. . . .

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To avoid losing think of another very well known game where the patterns of play are similar.

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A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .

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A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?

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Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo lives?

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Can you mark 4 points on a flat surface so that there are only two different distances between them?

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Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

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Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

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Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to. . . .

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We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

This is the first article in a series which aim to provide some insight into the way spatial thinking develops in children, and draw on a range of reported research. The focus of this article is the. . . .

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

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Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?

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Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

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Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

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There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you. . . .

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If you move the tiles around, can you make squares with different coloured edges?

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Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.

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A Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start. How many Hamiltonian circuits can you find in these graphs?

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How many moves does it take to swap over some red and blue frogs? Do you have a method?

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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?