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

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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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Is it possible to draw a 5-pointed star without taking your pencil off the paper? Is it possible to draw a 6-pointed star in the same way without taking your pen off?

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

This article for teachers describes how number arrays can be a useful representation for many number concepts.

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In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

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

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Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

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Explore the relationship between simple linear functions and their graphs.

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The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

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The number 8888...88M9999...99 is divisible by 7 and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M?

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

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A game for 2 people using a pack of cards Turn over 2 cards and try to make an odd number or a multiple of 3.

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Using the digits 1, 2, 3, 4, 5, 6, 7 and 8, mulitply a two two digit numbers are multiplied to give a four digit number, so that the expression is correct. How many different solutions can you find?

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

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Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?

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

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Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

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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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Benâ€™s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

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

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How can you use just one weighing to find out which box contains the lighter ten coins out of the ten boxes?

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Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?

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

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Have a go at balancing this equation. Can you find different ways of doing it?

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Can you work out some different ways to balance this equation?

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Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?

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

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What is the smallest number of answers you need to reveal in order to work out the missing headers?

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Does this 'trick' for calculating multiples of 11 always work? Why or why not?

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Find the highest power of 11 that will divide into 1000! exactly.

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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 different ways of creating paths using these paving slabs?

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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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An investigation that gives you the opportunity to make and justify predictions.

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These red, yellow and blue spinners were each spun 45 times in total. Can you work out which numbers are on each spinner?

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Find the words hidden inside each of the circles by counting around a certain number of spaces to find each letter in turn.

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The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

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

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This big box multiplies anything that goes inside it by the same number. If you know the numbers that come out, what multiplication might be going on in the box?