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?
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?
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.
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 find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
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?
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 different symmetrical shapes can you make by shading triangles or squares?
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?
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?
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.
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?
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?
Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?
Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.
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?
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.
Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.
Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.
Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?
An investigation that gives you the opportunity to make and justify predictions.
If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?
Pentagram Pylons - can you elegantly recreate them? Or, the European flag in LOGO - what poses the greater problem?
Six friends sat around a circular table. Can you work out from the information who sat where and what their profession were?
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.
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?
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
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?
These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.
Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.
Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.
When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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...
The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
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?
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?