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

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

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

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

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?

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?

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

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

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

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

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

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.

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.

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

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

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.

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

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

These tasks give learners chance to generalise, which involves identifying an underlying structure.

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 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.

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

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

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.

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

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

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?

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

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

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?

Can you find the values at the vertices when you know the values on the edges?

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

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.

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?

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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.

This article for primary teachers discusses how we can help learners generalise and prove, using NRICH tasks as examples.

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

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

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

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

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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

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?

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

This activity involves rounding four-digit numbers to the nearest thousand.

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

Try out this number trick. What happens with different starting numbers? What do you notice?