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?

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?

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?

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 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 challenging activity focusing on finding all possible ways of stacking rods.

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.

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?

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?

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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. . . .

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

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

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?

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

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

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?

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

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

Can you find all the different ways of lining up these Cuisenaire rods?

An activity making various patterns with 2 x 1 rectangular tiles.

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?

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.

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

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?

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

These practical challenges are all about making a 'tray' and covering it with paper.

In how many ways can you stack these rods, following the rules?

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.

Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.

Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

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

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.