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.

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

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?

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

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

Different combinations of the weights available allow you to make different totals. Which totals can you make?

Countries from across the world competed in a sports tournament. Can you devise an efficient strategy to work out the order in which they finished?

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

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.

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?

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

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.

Special clue numbers related to the difference between numbers in two adjacent cells and values of the stars in the "constellation" make this a doubly interesting problem.

Starting with four different triangles, imagine you have an unlimited number of each type. How many different tetrahedra can you make? Convince us you have found them all.

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

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?

An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

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

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

Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.

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?

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

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

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?

Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

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 have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?

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?

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?

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

A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

Four numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this sudoku.

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

A Sudoku with clues given as sums of entries.

A man has 5 coins in his pocket. Given the clues, can you work out what the coins are?

You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

This is a variation of sudoku which contains a set of special clue-numbers. Each set of 4 small digits stands for the numbers in the four cells of the grid adjacent to this set.

The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.