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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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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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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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Use your knowledge of place value to try to win this game. How will you maximise your score?

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Can you find different ways of creating paths using these paving slabs?

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In this simulation of a balance, you can drag numbers and parts of number sentences on to the trays. Have a play!

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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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Use your logical reasoning to work out how many cows and how many sheep there are in each field.

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Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?

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Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

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

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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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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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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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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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A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.

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What happens when you add three numbers together? Will your answer be odd or even? How do you know?

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Three dice are placed in a row. Find a way to turn each one so that the three numbers on top of the dice total the same as the three numbers on the front of the dice. Can you find all the ways to do. . . .

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Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?

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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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Are these statements always true, sometimes true or never true?

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Without doing lots of calculations, can you decide which of these number sentences are true? How do you know?

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

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Max and Bryony both have a box of sweets. What do you know about the number of sweets they each have?

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

In this article for primary teachers we consider in depth when we might reason which helps us understand what reasoning 'looks like'.

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There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.

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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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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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Are these statements relating to odd and even numbers always true, sometimes true or never true?

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Baker, Cooper, Jones and Smith are four people whose occupations are teacher, welder, mechanic and programmer, but not necessarily in that order. What is each person’s occupation?

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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.

This article for primary teachers suggests ways in which we can help learners move from being novice reasoners to expert reasoners.

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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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Can you find out which 3D shape your partner has chosen before they work out your shape?

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Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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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.

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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Are these statements always true, sometimes true or never true?

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

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In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

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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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Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry