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?

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?

A game for 1 person. Can you work out how the dice must be rolled from the start position to the finish? Play on line.

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

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?

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

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

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?

In this article for primary teachers, Fran describes her passion for paper folding as a springboard for mathematics.

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

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

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

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

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

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 will you go about finding all the jigsaw pieces that have one peg and one hole?

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?

Move just three of the circles so that the triangle faces in the opposite direction.

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Can you find ways of joining cubes together so that 28 faces are visible?

What is the greatest number of squares you can make by overlapping three squares?

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

Eight children each had a cube made from modelling clay. They cut them into four pieces which were all exactly the same shape and size. Whose pieces are the same? Can you decide who made each set?

Which of these dice are right-handed and which are left-handed?

If you can post the triangle with either the blue or yellow colour face up, how many ways can it be posted altogether?

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

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?

A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

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?

A game for 2 players. Given a board of dots in a grid pattern, players take turns drawing a line by connecting 2 adjacent dots. Your goal is to complete more squares than your opponent.

Imagine a 3 by 3 by 3 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will have holes drilled through them?

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

Can you find a way of counting the spheres in these arrangements?

An extension of noughts and crosses in which the grid is enlarged and the length of the winning line can to altered to 3, 4 or 5.

This second article in the series refers to research about levels of development of spatial thinking and the possible influence of instruction.

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

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?

Make one big triangle so the numbers that touch on the small triangles add to 10.

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

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

Try to picture these buildings of cubes in your head. Can you make them to check whether you had imagined them correctly?

Take it in turns to place a domino on the grid. One to be placed horizontally and the other vertically. Can you make it impossible for your opponent to play?