I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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

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

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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?

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

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

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

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?

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

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?

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

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

Can you explain the strategy for winning this game with any target?

Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

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.

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.

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

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?

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

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. . . .

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

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.

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. . . .

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

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

Alf Coles writes about how he tries to create 'spaces for exploration' for the students in his classrooms.

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. . . .

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

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?

Got It game for an adult and child. How can you play so that you know you will always win?

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

Can you figure out how sequences of beach huts are generated?

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

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

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

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

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

Nim-7 game for an adult and child. Who will be the one 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.

Investigate the different ways that fifteen schools could have given money in a charity fundraiser.

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

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

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