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

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

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

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

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

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?

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

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

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

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

There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?

Can you order the digits from 1-6 to make a number which is divisible by 6 so when the last digit is removed it becomes a 5-figure number divisible by 5, and so on?

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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

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.

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

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...

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?

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 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?

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

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.

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

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

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?

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

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.

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?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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?

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.

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?

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?

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.

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

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

George and Jim want to buy a chocolate bar. George needs 2p more and Jim need 50p more to buy it. How much is the chocolate bar?

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

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?

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.

Use the differences to find the solution to this Sudoku.

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?

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

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?

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

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

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. . . .