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?

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 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?

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

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?

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

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

Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

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

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

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?

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

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...

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

Your challenge is to find the longest way through the network following this rule. You can start and finish anywhere, and with any shape, as long as you follow the correct order.

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?

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

Find out what a "fault-free" rectangle is and try to make 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?

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.

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?

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.

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

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?

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

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?

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

What happens when you round these three-digit numbers to the nearest 100?

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?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

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

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

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

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

Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

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?

What happens when you round these numbers to the nearest whole number?

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?

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.

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

Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

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?