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.

What happens when you add three numbers together? Will your answer be odd or even? How do you know?

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

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

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?

A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.

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

Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

This Sudoku requires you to do some working backwards before working forwards.

You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?

Add or subtract the two numbers on the spinners and try to complete a row of three. Are there some numbers that are good to aim for?

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

Find the values of the nine letters in the sum: FOOT + BALL = GAME

If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?

Four small numbers give the clue to the contents of the four surrounding cells.

Throughout these challenges, the touching faces of any adjacent dice must have the same number. Can you find a way of making the total on the top come to each number from 11 to 18 inclusive?

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

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.

The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?

How many solutions can you find to this sum? Each of the different letters stands for a different number.

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

Find out about Magic Squares in this article written for students. Why are they magic?!

Replace the letters with numbers to make the addition work out correctly. R E A D + T H I S = P A G E

An investigation that gives you the opportunity to make and justify predictions.

This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?

Given the products of adjacent cells, can you complete this Sudoku?

This Sudoku, based on differences. Using the one clue number can you find the solution?

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

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

Given the products of diagonally opposite cells - can you complete this Sudoku?

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

A monkey with peaches, keeps a fraction of them each day, gives the rest away, and then eats one. How long can his peaches last?

A Sudoku that uses transformations as supporting clues.

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

The clues for this Sudoku are the product of the numbers in adjacent squares.

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

The puzzle can be solved with the help of small clue-numbers which are either placed on the border lines between selected pairs of neighbouring squares of the grid or placed after slash marks on. . . .

Using the statements, can you work out how many of each type of rabbit there are in these pens?

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.