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?

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?

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?

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

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

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

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?

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?

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

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?

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?

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?

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?

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

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

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

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

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

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

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

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

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

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

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

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?

What is the least number of moves you can take to rearrange the bears so that no bear is next to a bear of the same colour?

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

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

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

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

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

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

How many balls of modelling clay and how many straws does it take to make these skeleton shapes?

Exploring and predicting folding, cutting and punching holes and making spirals.

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

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?

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

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?

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

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

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

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?

Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?