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. . . .

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

Find your way through the grid starting at 2 and following these operations. What number do you end on?

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

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

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?

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

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

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

El Crico the cricket has to cross a square patio to get home. He can jump the length of one tile, two tiles and three tiles. Can you find a path that would get El Crico home in three jumps?

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

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

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

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

When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

Winifred Wytsh bought a box each of jelly babies, milk jelly bears, yellow jelly bees and jelly belly beans. In how many different ways could she make a jolly jelly feast with 32 legs?

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?

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

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

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

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?

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

Lorenzie was packing his bag for a school trip. He packed four shirts and three pairs of pants. "I will be able to have a different outfit each day", he said. How many days will Lorenzie be away?

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

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?

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

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

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?

My briefcase has a three-number combination lock, but I have forgotten the combination. I remember that there's a 3, a 5 and an 8. How many possible combinations are there to try?

Can you find out in which order the children are standing in this line?

Chandra, Jane, Terry and Harry ordered their lunches from the sandwich shop. Use the information below to find out who ordered each sandwich.

Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?

This challenge is about finding the difference between numbers which have the same tens digit.

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.

You have 5 darts and your target score is 44. How many different ways could you score 44?

Find all the numbers that can be made by adding the dots on two dice.

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

In Sam and Jill's garden there are two sorts of ladybirds with 7 spots or 4 spots. What numbers of total spots can you make?

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

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

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.

My coat has three buttons. How many ways can you find to do up all the buttons?

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

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

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?

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

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?

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?

The Red Express Train usually has five red carriages. How many ways can you find to add two blue carriages?