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?
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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.
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?
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?
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.
How many solutions can you find to this sum? Each of the different letters stands for a different number.
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
How many different symmetrical shapes can you make by shading triangles or squares?
Just four procedures were used to produce a design. How was it
done? Can you be systematic and elegant so that someone can follow
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?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
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.
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Pentagram Pylons - can you elegantly recreate them? Or, the
European flag in LOGO - what poses the greater problem?
Explore this how this program produces the sequences it does. What
are you controlling when you change the values of the variables?
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
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.
Find out about Magic Squares in this article written for students. Why are they magic?!
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?
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 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.
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
Remember that you want someone following behind you to see where
you went. Can yo work out how these patterns were created and
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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?
Hover your mouse over the counters to see which ones will be
removed. Click to remover them. The winner is the last one to
remove a counter. How you can make sure you win?
Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?
This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?
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.
A dog is looking for a good place to bury his bone. Can you work
out where he started and ended in each case? What possible routes
could he have taken?
Four small numbers give the clue to the contents of the four
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
A Sudoku with clues as ratios.
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?
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....?
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. . . .
A Sudoku with a twist.
A Latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and exactly once in each column.
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
How can you put five cereal packets together to make different
shapes if you must put them face-to-face?
60 pieces and a challenge. What can you make and how many of the
pieces can you use creating skeleton polyhedra?
Use the differences to find the solution to this Sudoku.
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?
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?
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 Sudoku that uses transformations as supporting clues.
This Sudoku puzzle can be solved with the help of small clue-numbers on the border lines between pairs of neighbouring squares of the grid.