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

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

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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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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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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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Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

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

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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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What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

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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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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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With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

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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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Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

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

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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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If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

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

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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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Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

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

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Explore the effect of reflecting in two intersecting mirror lines.

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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?

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

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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?

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What's the largest volume of box you can make from a square of paper?

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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

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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?

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

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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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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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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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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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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 little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

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

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Explore the effect of reflecting in two parallel mirror lines.

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.