Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
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?
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?
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 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 arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
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?
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?
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?
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.
How many different symmetrical shapes can you make by shading triangles or squares?
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?
The clues for this Sudoku are the product of the numbers in adjacent 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?
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.
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
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.
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 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?
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
Pentagram Pylons - can you elegantly recreate them? Or, the European flag in LOGO - what poses the greater problem?
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?
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...
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?
Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
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?
An investigation that gives you the opportunity to make and justify predictions.
Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.
The Zargoes use almost the same alphabet as English. What does this birthday message say?
Six friends sat around a circular table. Can you work out from the information who sat where and what their profession were?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Each of the main diagonals of this sudoku must contain the numbers 1 to 9 and each rectangle width the numbers 1 to 4.
60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?
What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?
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.
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?
Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?
Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.
Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?
You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.
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.
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?
This sudoku requires you to have "double vision" - two Sudoku's for the price of one