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?

Arrange 3 red, 3 blue and 3 yellow counters into a three-by-three square grid, so that there is only one of each colour in every row and every column

Jack has nine tiles. He put them together to make a square so that two tiles of the same colour were not beside each other. Can you find another way to do it?

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

The brown frog and green frog want to swap places without getting wet. They can hop onto a lily pad next to them, or hop over each other. How could they do it?

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

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?

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

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

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?

How many different shapes can you make by putting four right- angled isosceles triangles together?

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

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

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

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

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?

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 information to describe these marbles. What colours must be on marbles that sparkle when rolling but are dark inside?

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

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?

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?

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?

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?

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

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

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

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?

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?

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

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

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?

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

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

These eleven shapes each stand for a different number. Can you use the multiplication sums to work out what they are?

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.

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

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!

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?

What do you notice about the date 03.06.09? Or 08.01.09? This challenge invites you to investigate some interesting dates yourself.

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?

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

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?

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?

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?

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

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!

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?

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?