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?

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?

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?

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

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?

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?

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?

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?

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

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

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

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

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.?

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

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

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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?

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

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

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?

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

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

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?

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?

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

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

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.

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

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

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

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

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

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?

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?

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!

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?

Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

Tim's class collected data about all their pets. Can you put the animal names under each column in the block graph using the information?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

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?

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?

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?

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