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?

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

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?

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

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

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

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.

Take three differently coloured blocks - maybe red, yellow and blue. Make a tower using one of each colour. How many different towers can you make?

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 models can you find which obey these rules?

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

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?

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?

How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

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?

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.

If you had 36 cubes, what different cuboids could you make?

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?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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

Can you find out in which order the children are standing in this line?

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

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

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

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?

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

Start with three pairs of socks. Now mix them up so that no mismatched pair is the same as another mismatched pair. Is there more than one way to do it?

Can you fill in the empty boxes in the grid with the right shape and colour?

How many trains can you make which are the same length as Matt's and Katie's, using rods that are identical?

El Crico the cricket has to cross a square patio to get home. He can jump the length of one tile, two tiles and three tiles. Can you find a path that would get El Crico home in three jumps?

A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

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

Find all the numbers that can be made by adding the dots on two dice.

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

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. . . .

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

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

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make?

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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?