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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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Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

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Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . .

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Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . .

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How good are you at finding the formula for a number pattern ?

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Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

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Choose any four consecutive even numbers. Multiply the two middle numbers together. Multiply the first and last numbers. Now subtract your second answer from the first. Try it with your own. . . .

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Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

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A job needs three men but in fact six people do it. When it is finished they are all paid the same. How much was paid in total, and much does each man get if the money is shared as Fred suggests?

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Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

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A task which depends on members of the group noticing the needs of others and responding.

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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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Make some loops out of regular hexagons. What rules can you discover?

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My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

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If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?

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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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Can you see how to build a harmonic triangle? Can you work out the next two rows?

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Surprising numerical patterns can be explained using algebra and diagrams...

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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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Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?

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Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

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15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

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Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?

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Can you find rectangles where the value of the area is the same as the value of the perimeter?

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Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?

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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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Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .

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

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Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

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Can you explain what is going on in these puzzling number tricks?

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Can you explain why a sequence of operations always gives you perfect squares?

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The Number Jumbler can always work out your chosen symbol. Can you work out how?

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How many winning lines can you make in a three-dimensional version of noughts and crosses?

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Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

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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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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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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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Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

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

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32 x 38 = 30 x 40 + 2 x 8; 34 x 36 = 30 x 40 + 4 x 6; 56 x 54 = 50 x 60 + 6 x 4; 73 x 77 = 70 x 80 + 3 x 7 Verify and generalise if possible.

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Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

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I added together the first 'n' positive integers and found that my answer was a 3 digit number in which all the digits were the same...

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