Find out about Magic Squares in this article written for students. Why are they magic?!
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?
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?
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?
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?
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.
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?
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?
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.
A Sudoku with a twist.
Bellringers have a special way to write down the patterns they
ring. Learn about these patterns and draw some of your own.
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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 arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
Investigate the different ways you could split up these rooms so
that you have double the number.
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 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?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Solve this Sudoku puzzle whose clues are in the form of sums of the
numbers which should appear in diagonal opposite cells.
A Sudoku with clues as ratios.
How many different symmetrical shapes can you make by shading triangles or squares?
Investigate all the different squares you can make on this 5 by 5
grid by making your starting side go from the bottom left hand
point. Can you find out the areas of all these squares?
Two sudokus in one. Challenge yourself to make the necessary
What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Four numbers on an intersection that need to be placed in the
surrounding cells. That is all you need to know to solve this
Here you see the front and back views of a dodecahedron. Each
vertex has been numbered so that the numbers around each pentagonal
face add up to 65. Can you find all the missing numbers?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Find out what a "fault-free" rectangle is and try to make some of
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?
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.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.
How many models can you find which obey these rules?
Can you coach your rowing eight to win?
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?
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?
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?
A Sudoku with clues as ratios or fractions.
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 sudoku requires you to have "double vision" - two Sudoku's for
the price of one
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
A Sudoku that uses transformations as supporting clues.