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

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?

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?

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?

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

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

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

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

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

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?

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.

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

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

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

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?

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?

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?

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

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?

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

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

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

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?

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....?

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

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

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

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

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

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?

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?

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

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

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

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?

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

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

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?

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?

Use the information to describe these marbles. What colours must be on marbles that sparkle when rolling but are dark inside?

Find out what a "fault-free" rectangle is and try to make some of your own.

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?