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An exploration and challenge with some less usual cubes

A number exploration that gives the opportunity to develop a systematic approach.

An exploration involving number sequences and numbers associated with shapes - triangles, squares, pentagons etc.

Ever played a game of trying to draw a pattern without taking your pen off the paper?

Can you connect gas, water and electricity to the houses without any pipes crossing?

Given a network diagram showing a friendship group, can you work out who's who?

How do you measure the distance between two points? A straight line? The shortest route by road? The time it takes to travel? Find out different ways of measuring distance and the mathematical idea of a metric.

Can you find a winning strategy for Last Biscuit?

You can visualise triangular numbers as a triangular array of little squares... Can you use the image to prove some triangle number relationships?

Find out about centred shapes here! You can watch a short video that shows, in pictures, how centred triangles are made up of a triangular number and a square number. Then, why not explore some of these suggested number relationships yourself?

Grab a ruler, paper and scissors, construct a dodecagon, and cut it up. Can you reassemble it in a way to help you find its area?

If you have mastered Slippery Snail, come and take it further!

Can you beat the slippery snail spiral?

If you've enjoyed Approaching midnight, then go for the ultimate Approaching Midnight challenge!

Some help to get started on Approaching Midnight.

If you have enjoyed Approaching Midnight then explore these new variants!

Only take a peek here if you want to be shown how to fold the piece of paper in our Folding and Cutting task.

Here are some further tangram challenges for you to try before having a go at the main Rearrange and Make task.

What do you think of when you hear the word 'diamond'?

Take a look at how some students started working on our 'Fitted' challenge.

It's said that every picture tells a story... Can you tell the story of our video about Cube Numbers?

Explore these arrangements of circles. This task challenges you to consider what your idea of the 'next size up' might be.

What happens when you overlap pieces of paper? You might want to look at shape and areas. It's an exploration that can lead ... who knows where?

Have a go at arranging numbers in a special way to get a certain total. The adventure starts when you try to find all the possible ways of doing this - how will you know you haven't left out any possibilities?

Watch this video which shows you how to cut a snowflake from the six-fold paper template.

Watch someone making the six-fold paper template in preparation for creating a snowflake.

Can you work out how to make one of these paper snowflakes? You're likely to need to use what you know about symmetry as well as your visualisation skills.

Here are some simpler shapes to make by folding and cutting a single sheet of paper, which you could try before you tackle Make a Snowflake.

Can you remove the last bead from its string?

Join the countdown from seven to zero! Will you be the one to take the last counter and win the game?

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. But when they shared them with more friends, there were two left over...

We are used to calculating the mean, median and mode from a set of numbers, but can you find the numbers if you know the averages?

Can any shape with straight sides be made by folding a single piece of paper and by making a single complete cut?

How many different lengths is it possible to measure with a set of three rods? Explore this problem, and make some discoveries about number bases.

I wonder if there's a different way to climb 13 steps for each day of the year... Solving this problem might lead you to a famous number sequence.

Can you be the one to remove the last drip?

How many rectangles are there on a counting stick? How can you be sure you haven't missed any?

Can you be the first to hit 23? Play the game and see if you can come up with a winning strategy.

There are two ways to join three squares together edge to edge. What can you discover about the L-shaped triomino?

Perhaps you'd like to try making this Jackson cube?

We started drawing some quadrilaterals - can you complete them?

Can you use a picture or diagram to help Charlie and Liz decide whether they will be able to share their squares of chocolate equally?

Put a counter on a 100-square, and then move it around using only two operations. Which numbers can you visit?

Think of a two digit number, reverse the digits, and add the numbers together. Something special happens... Watch the videos and adapt the method to explain a variety of number tricks.

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

Egyptian fractions are sums of unit fractions. We invite you to explore ways of expressing a fraction as an Egyptian Fraction.

Think of a number, square it and subtract your starting number. Is there something special about the number you are left with?

Investigate the number of squares or lines that make up these spirals as they grow. This task encourages you to create new spirals of your own to explore in similar ways.

Can you create each of the letters in your name from a piece of paper by folding and making just one cut? Intrigued?

Can you really create a square from all these identical triangles? You may need to think about properties of 2D shapes. Don't give up!

Draw some shapes on a dotty grid, and explore the relationships between the dots on the edge, the dots inside, and the area. Is there a theorem to describe what's happening?

Can you find a quick way to determine whether four coordinates are the corners of a square?

Can you create a complete square before the computer? Play the interactive game and see if you can find a winning strategy.

In a world with just pennies, your purse or pockets would be very heavy. Two-coin systems make life much easier... Which coins would you choose?

If every seat in the cinema is filled, and you know how much money was taken, can you work out who was in the audience? A chance to explore some simple examples of Diophantine Equations.

Investigate the paths taken by a squirrel travelling to the corners of a quadrilateral but only going half-way each time. A wild exploration that may involve direction, area, length and shape!

An exploration involving quarters of shapes. You could take a look at some ways in which other people have been creative in response to this task!

Win this game by escaping the deathly vortex!

Can you find many different ways to write a fraction as an Egyptian Fraction? And what does it have to do with factors?

Alison introduces Charlie and Liz to a new kind of chocolate bar. How could you use an image to decide when they will be able to share their squares of chocolate equally?

What is special about numbers that can be written as the difference of two squares?

Play with our interactive machine and see if you can work out how to switch each light on. Is there a way to turn on all four lights at once?

Each number must be a factor or multiple of the last. What's the longest chain you can make?

Imagine you have a large supply of 3kg and 8kg weights. This problem invites you to explore the means of different sets of these weights.

Create your own grid of numbers following these simple rules. What fascinating patterns and connections can you find in the numbers? Be prepared to discover something new and perhaps surprising!

Have a go at making this skeletal octahdron.

A rectangle of paper, as many folds as you like and just one cut - can you create each diamond playing card in this way? Come on, have a go!

Can you fit all nine squares together to make a rectangle? Explore this intriguing task which can be tackled in a variety of ways.

To get around a city, we can't walk as the crow flies, but instead we have to walk up, down and across the blocks. What can you discover about taxi-cab geometry?

Watch our videos to see how we measured different lengths with sets of rods. Then explore which sets of rods are best for measuring longer and longer lengths...

Catch the last drop and win!

Race the clock to the midnight hour!

Or why not try your hand at creating a tetrahedron?

Once you can write unit fractions as Egyptian Fractions, you might like to try increasing the numerator...

When four friends meet, they decide to share their chocolate equally. When will this be possible? Can you create an image (diagram) that helps you decide?

Watch the video to see what Charlie can do with consecutive numbers. There are plenty of patterns to notice and explain...

Surely you can't cut up a playing card so that it's big enough to walk through ... Or can you?

Explore folding a sheet of paper and making just one cut. How many pieces of paper do you get? Can you make any number of pieces of paper?

Explore making squares of different sizes and all manner of other shapes using just the seven tangram pieces. You'll need to make use of your knowledge of properties of shapes and perhaps some visualisation.

Try to visualise the route taken by the arrows in our animation. What patterns do you notice?

If you wanted to draw squares on dotty paper, could you start with any two dots?

A chessboard has lots of squares... more than you might think at first... can you work out how many?

Make a splash with a game of Drips!

Explore how using a so-called Greedy Algorithm can help you to express different fractions as Egyptian Fractions. Will the method work for every fraction?

A piece of paper and a hole-punch will lead you into visualising, predicting, noticing and explaining.

This activity begins by inviting you to add pairs of consecutive numbers. What is the same about all the answers? Can you explain what is happening?

What happens when you add powers of numbers together? Charlie and Alison have been exploring. Can you explain their findings?

Explore the interactive number pyramid. Can you figure out how to make any number appear at the top of the pyramid?

Sometimes the cinema can sell every seat while taking exactly £100, in more than one way. Sometimes it can't be done at all! How do you know when it can and can't be done?

Mastered Got It? Then see where to go next!

Pentominoes are made by joining five squares together edge to edge. Can you find them all and make a full set?

Imagine you had a set of positive and negative "weights", which can be combined to make different totals. Which totals can you make?

Can you find sets of numbers whose mode, mean, median and range are all the same?

A robber is hiding at a crossroads in a modern city, where the roads are all at right angles and equally spaced.
Can you help the police to find him?

If you were given a diagonal of a square, do you think you could figure out where all four corners go?

If you wanted to design an efficient currency system, which coin denominations would you choose?

If we only had 3p and 7p coins, there'd be some values that we couldn't make. I wonder which ones? Explore this question and find out all about the Frobenius coin problem.

Can you find any other relationships between different centred shapes, triangular numbers and square numbers? Can you draw diagrams to show why the relationship is true?

Try covering a chessboard with L-triominoes. You can't cover every square... which squares can be left uncovered?

Can you find sets of numbers where mean < median < mode? Or median < mean < mode? Or...

Draw polygons on different sizes of dotty grid. What's the maximum number of sides you can create?

Watch the animation to see how to draw a mystic rose. Can you work out how many lines would be needed for roses of different sizes?

Choose a few numbers and add them up in pairs. You'll be surprised how much there is to discover...

Explore the properties of vectors and find out what happens when you use them to describe a journey round a square.

Charlie has 3 coins in his pocket that add up to 32p - what could they be? Come and explore this and other exciting questions about the maths of partitioning.

Can you cover a chessboard with dominoes? What if the chessboard has missing squares? Explore this famous puzzle and see if you can find an elegant solution.

I thought of 4 numbers - can you figure them out from 6 clues? Can you figure out 5 numbers from 10 clues? 6 numbers from 15 clues...

In this problem, explore the values you can make with different coins, learn about  Frobenius Numbers, and discover a way to work them out.

Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions...

Charlie has been colouring in square numbers and sums of squares on different grid. There are interesting patterns to explore and explain...

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. Try it a few times - notice anything?

Is there a quick way to work out the area of a square drawn on a dotty grid? What if the square is tilted?

If you wanted to draw a rhombus on dotty paper, could you start with any two dots?

Take a set of numbers and wipe one of them out. How does the average change? Can you do a little mathematical detective work to figure out which one? Can you develop a strategy?

Imagine you've won some coins from a slot machine. Can you find any patterns in the possible combinations of coins? Can you explain them?

Play with the interactivity and see what you notice. What interesting questions are there to ask about the diagonals of rectangles?

Can you design a good strategy for finding out from which floor you can drop an egg without it smashing?

Can you find rules for turning on pairs of lights, or even three or four lights?

Some shapes to make and explore with pentominoes

Can you draw some rhombuses that share a diagonal? There may be more than you are expecting...

If you want to know how many ways there are of making different amounts of money, you need a really good method which makes sure you don't miss any combinations!

Watch the video to see how Charlie and Becky make rectangles with squares, sticks and units. Then have a go at making some rectangles of your own...

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

In the video, Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can you figure out a way they could stop him?

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number? Explore this and other related questions, and see if you can prove your findings.

Cut up a square and rearrange the pieces to make a rectangle. The shapes have a different area - how can this be possible?

A parallelogram can be defined by two vectors. Can you discover a relationship between the two vectors and the area of the parallelogram?

In a three dimensional city on a distant planet, the metagiff has escaped from the zoo! Can you help the zookeeper locate the metagiff?

I wonder if there's a different way of feeding the car park machine for each day of the year... and what's it got to do with climbing stairs?

Is there a quick way to work out the number of different ways to go up staircases if I can take giant steps?

Complete the staircase to win the game!

Win this game by forcing your opponent to cross out the last dot!

If you're up for a real challenge, have a go at drawing some pictures to prove these relationships between centred shape numbers and other numbers.