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.

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

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.

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

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

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

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

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?

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

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

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

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 Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

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?

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

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

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

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?

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.

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.

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

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?

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

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

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

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?

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

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.

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

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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

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.

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

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

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

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?

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

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.

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?

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

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

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

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

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