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?

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.

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.

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

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?

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

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

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

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

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

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.

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?

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 problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

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

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?

This task follows on from Build it Up and takes the ideas into three dimensions!

Can you find all the ways to get 15 at the top of this triangle of numbers?

A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?"

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?

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

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.

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!

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.

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?

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?

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

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

This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.

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

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

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

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?

There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?

You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

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?

Given the products of adjacent 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?

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?

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

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

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

Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

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

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?

There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places.

You have 5 darts and your target score is 44. How many different ways could you score 44?