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

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?

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

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....?

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

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?

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

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

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

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

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

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?

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

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

Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?

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

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

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

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

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

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

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.

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?

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

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

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

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

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?

Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?

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

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.

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.

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?

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

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

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?

In this matching game, you have to decide how long different events take.

In this maze of hexagons, you start in the centre at 0. The next hexagon must be a multiple of 2 and the next a multiple of 5. What are the possible paths you could take?

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?

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.

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

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

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