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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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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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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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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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Can you produce convincing arguments that a selection of statements about numbers 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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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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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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Can you explain why a sequence of operations always gives you perfect squares?

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

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

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

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

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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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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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We have exactly 100 coins. There are five different values of coins. We have decided to buy a piece of computer software for 39.75. We have the correct money, not a penny more, not a penny less! Can. . . .

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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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Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

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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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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What is the largest number of intersection points that a triangle and a quadrilateral can have?

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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?

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

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

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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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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?