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.

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

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

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?

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

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

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

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

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.

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

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?

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

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?

For this challenge, you'll need to play Got It! 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.

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

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?

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

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

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?

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

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

Are these statements always true, sometimes true or never true?

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

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?

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?

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

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

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.

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

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.

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 you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

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?

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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

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?

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

Find out what a "fault-free" rectangle is and try to make some of your own.

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.

How many centimetres of rope will I need to make another mat just like the one I have here?

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