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

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

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

How many moves does it take to swap over some red and blue frogs? Do you have a method?

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.

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

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. . . .

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

Make some loops out of regular hexagons. What rules can you discover?

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

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

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

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

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

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?

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

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

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?

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

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

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

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.

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

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?

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

Can you work out how to win this game of Nim? Does it matter if you go first or second?

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

Explore the effect of reflecting in two intersecting mirror lines.

Explore the effect of combining enlargements.

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. . . .

To avoid losing think of another very well known game where the patterns of play are similar.

In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

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.

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.