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

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?

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?

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. . . .

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?

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?

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

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?

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

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

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?

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?

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?

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

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?

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 can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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

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?

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

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 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?

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

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 toy has a regular tetrahedron, a cube and a base with triangular and square hollows. If you fit a shape into the correct hollow a bell rings. How many times does the bell ring in a complete game?

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?

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?

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

Can you find ways of joining cubes together so that 28 faces are visible?

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

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?

How many different symmetrical shapes can you make by shading triangles or squares?

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?

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 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?

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?

When dice land edge-up, we usually roll again. But what if we didn't...?

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

The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .

Make a cube out of straws and have a go at this practical challenge.

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

A shape and space game for 2,3 or 4 players. Be the last person to be able to place a pentomino piece on the playing board. Play with card, or on the computer.

An extension of noughts and crosses in which the grid is enlarged and the length of the winning line can to altered to 3, 4 or 5.

A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.