Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

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

Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.

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

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 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 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.

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

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

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

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

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

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

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 loser is the player who takes the last counter.

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

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

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.

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.

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.

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.

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

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.

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

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

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

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.

A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand. . . .

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

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

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

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?

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

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.

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

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

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.

To avoid losing think of another very well known game where the patterns of play are similar.

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?

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?

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

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?