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

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

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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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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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Which set of numbers that add to 10 have the largest product?

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

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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 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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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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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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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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Choose any three by three square of dates on a calendar page...

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

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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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Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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

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You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

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How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

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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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Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

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Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

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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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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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I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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Can you explain why a sequence of operations always gives you perfect squares?

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You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?

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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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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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Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

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From a group of any 4 students in a class of 30, each has exchanged Christmas cards with the other three. Show that some students have exchanged cards with all the other students in the class. How. . . .

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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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Four jewellers share their stock. Can you work out the relative values of their gems?

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

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

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Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?

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I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

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I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

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Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?

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