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

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

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

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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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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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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.

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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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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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?

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

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

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

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

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

This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .

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

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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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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Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

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

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

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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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In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

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Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to. . . .

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Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

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

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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.