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Substitution and Transposition all in one! How fiendish can these codes get?

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Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

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Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?

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Nearly all of us have made table patterns on hundred squares, that is 10 by 10 grids. This problem looks at the patterns on differently sized square grids.

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48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?

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Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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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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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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Can you see how these factor-multiple chains work? Find the chain which contains the smallest possible numbers. How about the largest possible numbers?

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Can you make square numbers by adding two prime numbers together?

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Your vessel, the Starship Diophantus, has become damaged in deep space. Can you use your knowledge of times tables and some lightning reflexes to survive?

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56 406 is the product of two consecutive numbers. What are these two numbers?

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On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?

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

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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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One quarter of these coins are heads but when I turn over two coins, one third are heads. How many coins are there?

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How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

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Work out Tom's number from the answers he gives his friend. He will only answer 'yes' or 'no'.

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Watch the video of this game being played. Can you work out the rules? Which dice totals are good to get, and why?

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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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"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?

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The discs for this game are kept in a flat square box with a square hole for each. Use the information to find out how many discs of each colour there are in the box.

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Here is a chance to create some Celtic knots and explore the mathematics behind them.

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

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Number problems at primary level to work on with others.

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Does a graph of the triangular numbers cross a graph of the six times table? If so, where? Will a graph of the square numbers cross the times table too?

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Number problems at primary level that may require resilience.

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What is the lowest number which always leaves a remainder of 1 when divided by each of the numbers from 2 to 10?

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The items in the shopping basket add and multiply to give the same amount. What could their prices be?

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I throw three dice and get 5, 3 and 2. Add the scores on the three dice. What do you get? Now multiply the scores. What do you notice?

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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?

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Guess the Dominoes for child and adult. Work out which domino your partner has chosen by asking good questions.

This article for primary teachers outlines why developing an intuitive 'feel' for numbers matters, and how our activities focusing on factors and multiples can help.

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Can you complete this jigsaw of the multiplication square?

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What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

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

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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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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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In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?

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

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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?