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

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.

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?

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?

My cousin was 24 years old on Friday April 5th in 1974. On what day of the week was she born?

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?

You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

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

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

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

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

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

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.?

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

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

These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.

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

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

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

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

There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

Can you use this information to work out Charlie's house number?

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

Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

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

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

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

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?

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.

Six friends sat around a circular table. Can you work out from the information who sat where and what their profession were?

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

Tim's class collected data about all their pets. Can you put the animal names under each column in the block graph using the information?

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

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

In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?

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?

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.

Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

What is the smallest number of coins needed to make up 12 dollars and 83 cents?

I was in my car when I noticed a line of four cars on the lane next to me with number plates starting and ending with J, K, L and M. What order were they in?

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

These practical challenges are all about making a 'tray' and covering it with paper.

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

Investigate the different ways you could split up these rooms so that you have double the number.

Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.