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

Take three differently coloured blocks - maybe red, yellow and blue. Make a tower using one of each colour. How many different towers can you make?

Penta people, the Pentominoes, always build their houses from five square rooms. I wonder how many different Penta homes you can create?

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

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.

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

Can you find the chosen number from the grid using the clues?

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?

Start with three pairs of socks. Now mix them up so that no mismatched pair is the same as another mismatched pair. Is there more than one way to do it?

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?

Can you order the digits from 1-6 to make a number which is divisible by 6 so when the last digit is removed it becomes a 5-figure number divisible by 5, and so on?

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

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?

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

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?

Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number.

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

Imagine that the puzzle pieces of a jigsaw are roughly a rectangular shape and all the same size. How many different puzzle pieces could there be?

In this maze of hexagons, you start in the centre at 0. The next hexagon must be a multiple of 2 and the next a multiple of 5. What are the possible paths you could take?

These two group activities use mathematical reasoning - one is numerical, one geometric.

Can you make square numbers by adding two prime numbers together?

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

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!

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?

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?

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

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?

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?

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.

In how many ways could Mrs Beeswax put ten coins into her three puddings so that each pudding ended up with at least two coins?

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?

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

When you throw two regular, six-faced dice you have more chance of getting one particular result than any other. What result would that be? Why is this?

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

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.

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.

Moira is late for school. What is the shortest route she can take from the school gates to the entrance?

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make?

In this problem it is not the squares that jump, you do the jumping! The idea is to go round the track in as few jumps as possible.

Using the statements, can you work out how many of each type of rabbit there are in these pens?

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 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.

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?