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.

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

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

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

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

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

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

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.

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

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.

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

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

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.

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

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.

An investigation that gives you the opportunity to make and justify predictions.

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

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

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?

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

Delight your friends with this cunning trick! Can you explain how it works?

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

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

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

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

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.

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

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

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.

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.

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

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?

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

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? Many opportunities to work in different ways.

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?

Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

Watch this animation. What do you see? Can you explain why this happens?

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

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

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?

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

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