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.

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

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.

A game for 2 players with similarities 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.

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

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

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

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

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.

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

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

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.

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.

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

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?

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

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

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?

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

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

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

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

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

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

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

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

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

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

Here are two kinds of spirals for you to explore. What do you notice?

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

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

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?

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

This challenge asks you to imagine a snake coiling on itself.

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

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

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

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.

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

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

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

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?

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

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?

Can you find a way of counting the spheres in these arrangements?

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.