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However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

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

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

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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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If a sum invested gains 10% each year how long before it has doubled its value?

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The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

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 heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?

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

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

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

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

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

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The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

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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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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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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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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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The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

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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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Pick a square within a multiplication square and add the numbers on each diagonal. 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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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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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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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 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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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 � 1 [1/3]. What other numbers have the sum equal to the product and can this be so. . . .

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

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

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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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Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

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

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

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

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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.