Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

Explore the effect of reflecting in two intersecting mirror lines.

Explore the effect of reflecting in two parallel mirror lines.

Can you find the values at the vertices when you know the values on the edges?

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.

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.

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

Can you find sets of sloping lines that enclose a square?

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Can you describe this route to infinity? Where will the arrows take you next?

It starts quite simple but great opportunities for number discoveries and patterns!

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

Can all unit fractions be written as the sum of two unit fractions?

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

It would be nice to have a strategy for disentangling any tangled ropes...

Charlie has moved between countries and the average income of both has increased. How can this be so?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?