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The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

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List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

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A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

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A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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Can you explain the strategy for winning this game with any target?

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Got It game for an adult and child. How can you play so that you know you will always win?

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Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

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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 pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?

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I added together the first 'n' positive integers and found that my answer was a 3 digit number in which all the digits were the same...

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A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

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Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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Is there an efficient way to work out how many factors a large number has?

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Follow this recipe for sieving numbers and see what interesting patterns emerge.

Can you find any perfect numbers? Read this article to find out more...

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

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Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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You'll need to know your number properties to win a game of Statement Snap...

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Given the products of diagonally opposite cells - can you complete this Sudoku?

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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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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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Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

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Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

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How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?

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Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

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Given the products of adjacent cells, can you complete this Sudoku?

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Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

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Each letter represents a different positive digit AHHAAH / JOKE = HA What are the values of each of the letters?

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Play this game and see if you can figure out the computer's chosen number.

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Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

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The clues for this Sudoku are the product of the numbers in adjacent squares.

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Data is sent in chunks of two different sizes - a yellow chunk has 5 characters and a blue chunk has 9 characters. A data slot of size 31 cannot be exactly filled with a combination of yellow and. . . .

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The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.

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Nine squares are fitted together to form a rectangle. Can you find its dimensions?

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Explore the relationship between simple linear functions and their graphs.

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Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

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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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Factor track is not a race but a game of skill. The idea is to go round the track in as few moves as possible, keeping to the rules.

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Can you work out what size grid you need to read our secret message?

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Factors and Multiples game for an adult and child. How can you make sure you win this game?

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Each clue in this Sudoku is the product of the two numbers in adjacent cells.

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Can you find any two-digit numbers that satisfy all of these statements?

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Think of any three-digit number. Repeat the digits. The 6-digit number that you end up with is divisible by 91. Is this a coincidence?

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

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Complete the following expressions so that each one gives a four digit number as the product of two two digit numbers and uses the digits 1 to 8 once and only once.

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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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15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?