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?

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

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

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?

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.

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

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

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

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?

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

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.

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.

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?

Use the clues about the shaded areas to help solve this sudoku

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

Each of the main diagonals of this sudoku must contain the numbers 1 to 9 and each rectangle width the numbers 1 to 4.

A pair of Sudokus with lots in common. In fact they are the same problem but rearranged. Can you find how they relate to solve them both?

Solve the equations to identify the clue numbers in this Sudoku problem.

A Latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and exactly once in each column.

A pair of Sudoku puzzles that together lead to a complete solution.

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?

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

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

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?

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 Sudoku that uses transformations as supporting clues.

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?

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?

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.

Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?

In this Sudoku, there are three coloured "islands" in the 9x9 grid. Within each "island" EVERY group of nine cells that form a 3x3 square must contain the numbers 1 through 9.

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

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. . . .

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.

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?

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

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

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.

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

Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?

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

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