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Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

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Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

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

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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The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

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Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.

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

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First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

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ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

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When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

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It would be nice to have a strategy for disentangling any tangled ropes...

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It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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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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A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

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Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

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How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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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Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

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Can you describe this route to infinity? Where will the arrows take you next?

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Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

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Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

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Can you find sets of sloping lines that enclose a square?

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Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

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Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

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Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

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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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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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Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

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A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand. . . .

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A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.

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Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

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Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

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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.

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

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Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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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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Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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Nim-7 game for an adult and child. Who will be the one to take the last counter?

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It starts quite simple but great opportunities for number discoveries and patterns!