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?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

What is the best way to shunt these carriages so that each train can continue its journey?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?

My cousin was 24 years old on Friday April 5th in 1974. On what day of the week was she born?

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?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

How many different triangles can you make on a circular pegboard that has nine pegs?

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

These practical challenges are all about making a 'tray' and covering it with paper.

An activity making various patterns with 2 x 1 rectangular tiles.

What happens when you try and fit the triomino pieces into these two grids?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

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

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

How many different rhythms can you make by putting two drums on the wheel?

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?

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?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

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?

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?

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?

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

In how many ways can you stack these rods, following the rules?

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

Alice's mum needs to go to each child's house just once and then back home again. How many different routes are there? Use the information to find out how long each road is on the route she took.

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

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?

Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?

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?

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.