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.

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.

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.

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

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.

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. 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.

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?

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?

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

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?

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

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.

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

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.

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

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

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

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

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.

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

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

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

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

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

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

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

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

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

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

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?

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

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

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

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?

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

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

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

In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

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

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?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

What happens when you round these numbers to the nearest whole number?

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