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

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Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

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This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.

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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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Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .

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In the following sum the letters A, B, C, D, E and F stand for six distinct digits. Find all the ways of replacing the letters with digits so that the arithmetic is correct.

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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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You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

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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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Keep constructing triangles in the incircle of the previous triangle. What happens?

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

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Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

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Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

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Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

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The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .

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Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

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Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

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Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

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Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

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Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

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Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

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Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

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This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.

A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

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Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.

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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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Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

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Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?

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A huge wheel is rolling past your window. What do you see?

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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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The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

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

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How many noughts are at the end of these giant numbers?

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I am exactly n times my daughter's age. In m years I shall be ... How old am I?

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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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Prove Pythagoras' Theorem using enlargements and scale factors.

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A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

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Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

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Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

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Kyle and his teacher disagree about his test score - who is right?

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This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal. . . .

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The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.