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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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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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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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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.

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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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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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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?

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

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

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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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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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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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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.

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

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

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

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

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

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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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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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Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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Build gnomons that are related to the Fibonacci sequence and try to explain why this is 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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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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This task encourages you to investigate the number of edging pieces and panes in different sized windows.

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

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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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List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

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

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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 sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 ï¿½ 1 [1/3]. What other numbers have the sum equal to the product and can this be. . . .

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Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

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

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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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Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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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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Can you find the values at the vertices when you know the values on the edges?