Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?

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?

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

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

A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.

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?

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

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

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?

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

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

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

Seeing Squares game for an adult and child. Can you come up with a way of always winning this game?

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.

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

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?

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

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?

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?

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?

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?

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?

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?

This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .

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

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

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

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

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

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

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

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

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

What shape is made when you fold using this crease pattern? Can you make a ring design?

Here are shadows of some 3D shapes. What shapes could have made them?

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

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

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?

This article for teachers describes a project which explores the power of storytelling to convey concepts and ideas to children.

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.

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

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.

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?