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?

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?

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?

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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,. . . .

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?

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

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

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

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?

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?

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

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

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?

Can you fit the tangram pieces into the outline of the child walking home from school?

Can you fit the tangram pieces into the outlines of these clocks?

Can you fit the tangram pieces into the outlines of the chairs?

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?

Can you fit the tangram pieces into the outline of this shape. How would you describe it?

Can you fit the tangram pieces into the outlines of these people?

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?

Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?

Can you fit the tangram pieces into the outline of this telephone?

Given a 2 by 2 by 2 skeletal cube with one route `down' the cube. How many routes are there from A to B?

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

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

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

Exchange the positions of the two sets of counters in the least possible number of moves

Can you fit the tangram pieces into the outline of Little Fung at the table?

Can you fit the tangram pieces into the outline of Little Ming playing the board game?

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?

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

Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

Can you fit the tangram pieces into the outline of this goat and giraffe?

Can you fit the tangram pieces into the outline of this sports car?

Here's a simple way to make a Tangram without any measuring or ruling lines.

Can you fit the tangram pieces into the outline of Granma T?

Can you fit the tangram pieces into the outlines of the workmen?

Can you fit the tangram pieces into the outline of Little Ming and Little Fung dancing?

Can you fit the tangram pieces into the outline of these convex shapes?

Can you fit the tangram pieces into the outline of this junk?

Can you fit the tangram pieces into the outline of Wai Ping, Wah Ming and Chi Wing?

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