These rectangles have been torn. How many squares did each one have inside it before it was ripped?

How many ways can you find of tiling the square patio, using square tiles of different sizes?

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

Can you rearrange the biscuits on the plates so that the three biscuits on each plate are all different and there is no plate with two biscuits the same as two biscuits on another plate?

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

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

My cube has inky marks on each face. Can you find the route it has taken? What does each face look like?

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

These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and lollypops for 7p in the sweet shop. What could each of the children buy with their money?

Can you find which shapes you need to put into the grid to make the totals at the end of each row and the bottom of each column?

Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.

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?

Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?

This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.

What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?

How many trapeziums, of various sizes, are hidden in this picture?

The Zargoes use almost the same alphabet as English. What does this birthday message say?

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

What could the half time scores have been in these Olympic hockey matches?

Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?

Exactly 195 digits have been used to number the pages in a book. How many pages does the book have?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places.

On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

Using all ten cards from 0 to 9, rearrange them to make five prime numbers. Can you find any other ways of doing it?

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?

A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

On a digital clock showing 24 hour time, over a whole day, how many times does a 5 appear? Is it the same number for a 12 hour clock over a whole day?

In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!

There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

Stuart's watch loses two minutes every hour. Adam's watch gains one minute every hour. Use the information to work out what time (the real time) they arrived at the airport.

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?