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.

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

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?

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

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?

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

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

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

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

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.

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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

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

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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?

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?

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

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

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

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

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.

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?

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

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

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

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

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.

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

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

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

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

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.

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.

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.

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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

Can you describe this route to infinity? Where will the arrows take you next?

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

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

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

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.

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Charlie has moved between countries and the average income of both has increased. How can this be so?

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?

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