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?

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

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

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

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?

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?

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?

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

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

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?

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. . . .

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,. . . .

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

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?

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?

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

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?

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

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?

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?

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

We can cut a small triangle off the corner of a square and then fit the two pieces together. Can you work out how these shapes are made from the two pieces?

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

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.

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

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

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

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

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

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

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

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

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?

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?

Create a pattern on the left-hand grid. How could you extend your pattern on the right-hand grid?

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?

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?

This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.

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?

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?

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?

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?

Have you ever tried tessellating capital letters? Have a look at these examples and then try some for yourself.

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