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

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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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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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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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Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

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Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

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

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

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

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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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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 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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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?

This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.

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

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

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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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If you know the perimeter of a right angled triangle, what can you say about the area?

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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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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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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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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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What is special about the difference between squares of numbers adjacent to multiples of three?

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What is the total number of squares that can be made on a 5 by 5 geoboard?

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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. . . .