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

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

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?

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?

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?

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?

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?

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

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?

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

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?

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?

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. . . .

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?

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?

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

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

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

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

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

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?

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

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

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

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?

Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?

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?

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?

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?

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?

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'.

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

One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?

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

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

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?

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

Can you work out what is wrong with the cogs on a UK 2 pound coin?

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

What happens when you turn these cogs? Investigate the differences between turning two cogs of different sizes and two cogs which are the same.

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

This article looks at levels of geometric thinking and the types of activities required to develop this thinking.

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?

Can you fit the tangram pieces into the outlines of the convex shapes?