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

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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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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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Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

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

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

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Show that all pentagonal numbers are one third of a triangular 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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How good are you at finding the formula for a number pattern ?

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

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

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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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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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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 find a rule which connects consecutive triangular numbers?

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Can you find a rule which relates triangular numbers to square numbers?

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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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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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There are unexpected discoveries to be made about square numbers...

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

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A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

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

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What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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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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Play around with the Fibonacci sequence and discover some surprising results!

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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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Where should you start, if you want to finish back where you started?

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

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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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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. . . .