An investigation that gives you the opportunity to make and justify predictions.

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

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

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

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

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

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?

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

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

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

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

If these elves wear a different outfit every day for as many days as possible, how many days can their fun last?

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

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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

Complete the magic square using the numbers 1 to 25 once each. Each row, column and diagonal adds up to 65.

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?

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?

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?

The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

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

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

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

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

This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?

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

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.

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?

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

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

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

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?

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

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

This task follows on from Build it Up and takes the ideas into three dimensions!

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

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?

Can you find all the ways to get 15 at the top of this triangle of numbers?

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.

Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

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.

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?