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Visitors to Earth from the distant planet of Zub-Zorna were amazed when they found out that when the digits in this multiplication were reversed, the answer was the same! Find a way to explain. . . .

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Can you produce convincing arguments that a selection of statements about numbers are true?

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Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?

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115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

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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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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?

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

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

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The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

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How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?

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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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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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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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In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice?

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The number 27 is special because it is three times the sum of its digits 27 = 3 (2 + 7). Find some two digit numbers that are SEVEN times the sum of their digits (seven-up numbers)?

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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 the five distinct digits N, R, I, C and H in the following nomogram

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

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

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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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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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Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

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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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The sums of the squares of three related numbers is also a perfect square - can you explain why?

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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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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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Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?

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

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If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

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

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

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

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