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.

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?

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 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?

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?

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?

A Sudoku based on clues that give the differences between adjacent cells.

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.

Two sudokus in one. Challenge yourself to make the necessary connections.

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.

This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.

This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?

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

Two sudokus in one. Challenge yourself to make the necessary connections.

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?

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.

A Sudoku with clues given as sums of entries.

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?

A Sudoku that uses transformations as supporting clues.

This Sudoku puzzle can be solved with the help of small clue-numbers on the border lines between pairs of neighbouring squares of the grid.

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

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.

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

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

Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers?

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

Four friends must cross a bridge. How can they all cross it in just 17 minutes?

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

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?

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?

A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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

Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?

This sudoku requires you to have "double vision" - two Sudoku's for the price of one

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

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

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

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

Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar".

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

You need to find the values of the stars before you can apply normal Sudoku rules.

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

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

Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones.