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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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Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

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What is the best way to shunt these carriages so that each train can continue its journey?

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Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

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How will you go about finding all the jigsaw pieces that have one peg and one hole?

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Design an arrangement of display boards in the school hall which fits the requirements of different people.

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10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

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How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

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How many different triangles can you make on a circular pegboard that has nine pegs?

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Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

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In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

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This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

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Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

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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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Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

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How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

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Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

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Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

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Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

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Here are more buildings to picture in your mind's eye. Watch out - they become quite complicated!

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In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

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This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

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This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

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How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

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Can you find a way of counting the spheres in these arrangements?

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A game has a special dice with a colour spot on each face. These three pictures show different views of the same dice. What colour is opposite blue?

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Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

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A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

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Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?

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What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

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How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

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How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

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Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?

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Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

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The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .

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What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

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Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

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What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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Seeing Squares game for an adult and child. Can you come up with a way of always winning this game?

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The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

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A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

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

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Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?