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?

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

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?

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

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

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?

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

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

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?

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

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

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?

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

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?

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?

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

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

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 shapes can you make by putting four right- angled isosceles triangles together?

Can you find all the different triangles on these peg boards, and find their angles?

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

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 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?

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

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

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?

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

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

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

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

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

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

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

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

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. . . .

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.

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

My coat has three buttons. How many ways can you find to do up all the buttons?

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?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?