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.

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?

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?

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?

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?

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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?

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

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?

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

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

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?

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?

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

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

A few extra challenges set by some young NRICH members.

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

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?

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

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

Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

How many different symmetrical shapes can you make by shading triangles or squares?

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.

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

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

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

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?

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?

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

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

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.

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.

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?

If these elves wear a different outfit every day for as many days as possible, how many days can their fun last?

A package contains a set of resources designed to develop students’ mathematical thinking. This package places a particular emphasis on “being systematic” and is designed to meet. . . .

A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

When you throw two regular, six-faced dice you have more chance of getting one particular result than any other. What result would that be? Why is this?

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

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

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?

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

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

This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!

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

How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?

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.