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This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

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Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

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While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

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Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.

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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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Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?

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Three beads are threaded on a circular wire and are coloured either red or blue. Can you find all four different combinations?

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Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

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In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

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Your challenge is to find the longest way through the network following this rule. You can start and finish anywhere, and with any shape, as long as you follow the correct order.

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"Tell me the next two numbers in each of these seven minor spells", chanted the Mathemagician, "And the great spell will crumble away!" Can you help Anna and David break the spell?

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How many different sets of numbers with at least four members can you find in the numbers in this box?

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Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.

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EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.

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If the numbers 5, 7 and 4 go into this function machine, what numbers will come out?

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Investigate and explain the patterns that you see from recording just the units digits of numbers in the times tables.

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This number has 903 digits. What is the sum of all 903 digits?

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Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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Complete these two jigsaws then put one on top of the other. What happens when you add the 'touching' numbers? What happens when you change the position of the jigsaws?

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Investigate what happens when you add house numbers along a street in different ways.

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In this section from a calendar, put a square box around the 1st, 2nd, 8th and 9th. Add all the pairs of numbers. What do you notice about the answers?

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There are ten children in Becky's group. Can you find a set of numbers for each of them? Are there any other sets?

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This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

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Investigate the totals you get when adding numbers on the diagonal of this pattern in threes.

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Using only the red and white rods, how many different ways are there to make up the other colours of rod?

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Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

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Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?

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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?

These upper primary activities offer opportunities for children to recognise, extend and explain number patterns.

This article for primary teachers outlines how we can encourage children to create, identify, extend and explain number patterns and why being able to do so is useful.

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Investigate the successive areas of light blue in these diagrams.

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Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

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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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This activity creates an opportunity to explore all kinds of number-related patterns.

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Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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It starts quite simple but great opportunities for number discoveries and patterns!

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In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?

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Here are some ideas to try in the classroom for using counters to investigate number patterns.

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In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.

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If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?

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Investigate these hexagons drawn from different sized equilateral triangles.

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Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

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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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July 1st 2001 was on a Sunday. July 1st 2002 was on a Monday. When did July 1st fall on a Monday again?

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An environment which simulates working with Cuisenaire rods.

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How many moves does it take to swap over some red and blue frogs? Do you have a method?