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?

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?

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?

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?

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.

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 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?

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?

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?

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

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

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

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

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

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?

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

How many models can you find which obey these rules?

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

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

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

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

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?

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

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?

How many different rhythms can you make by putting two drums on the wheel?

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

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

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.

Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?

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

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.

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

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

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

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

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

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?

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?

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

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

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?

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....?

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.