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

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

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

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.

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

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?

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

A game for 2 players with similaritlies to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.

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

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

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. . . .

A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

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

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

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

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

Nim-7 game for an adult and child. Who will be the one to take the last counter?

Are these statements relating to odd and even numbers always true, sometimes true or never true?

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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.

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.

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

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

Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?

This task follows on from Build it Up and takes the ideas into three dimensions!

Can you find all the ways to get 15 at the top of this triangle of numbers?

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

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

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.

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.

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

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

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

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.

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

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