Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?
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.
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?
Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.
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.
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?
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 the different ways you could split up these rooms so that you have double the number.
If these elves wear a different outfit every day for as many days as possible, how many days can their fun last?
Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.
This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.
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?
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.
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.
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 tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.
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?
Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.
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?
Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?
Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.
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?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.
How many models can you find which obey these rules?
Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?
Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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.
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?
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
Penta people, the Pentominoes, always build their houses from five square rooms. I wonder how many different Penta homes you can create?
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?
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?
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?
Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.
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?
In how many ways can you stack these rods, following the rules?
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?
Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.
Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?
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?
Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?
These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.
These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.
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.
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
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.