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

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

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

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

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'.

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?

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?

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?

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

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

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. . . .

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

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

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?

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?

Can you make a 3x3 cube with these shapes made from small cubes?

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

An activity centred around observations of dots and how we visualise number arrangement patterns.

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

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 does the overlap of these two shapes look like? Try picturing it in your head and then use the interactivity to test your prediction.

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?

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

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

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,. . . .

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?

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?

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

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 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?

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

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 work out what shape is made when this piece of paper is folded up using the crease pattern shown?

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

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

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

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

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

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

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.

I found these clocks in the Arts Centre at the University of Warwick intriguing - do they really need four clocks and what times would be ambiguous with only two or three of them?

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

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.