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.
There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.
A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.
We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?
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?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Can you order pictures of the development of a frog from frogspawn
and of a bean seed growing into a plant?
My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?
If these elves wear a different outfit every day for as many days
as possible, how many days can their fun last?
Can you use the information to find out which cards I have used?
A Sudoku with clues given as sums of entries.
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.
Follow the clues to find the mystery number.
Can you see who the gold medal winner is? What about the silver medal winner and the bronze medal winner?
Find out about Magic Squares in this article written for students. Why are they magic?!
60 pieces and a challenge. What can you make and how many of the
pieces can you use creating skeleton polyhedra?
What happens when you add three numbers together? Will your answer be odd or even? How do you know?
This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?
Use the information to describe these marbles. What colours must be
on marbles that sparkle when rolling but are dark inside?
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 could the half time scores have been in these Olympic hockey matches?
Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.
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?
Solve this Sudoku puzzle whose clues are in the form of sums of the
numbers which should appear in diagonal opposite cells.
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.
Make your own double-sided magic square. But can you complete both
sides once you've made the pieces?
Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.
Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.
Make a pair of cubes that can be moved to show all the days of the
month from the 1st to the 31st.
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
In this article, the NRICH team describe the process of selecting solutions for publication on the site.
Can you find which shapes you need to put into the grid to make the
totals at the end of each row and the bottom of each column?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
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.
Arrange the digits 1, 1, 2, 2, 3 and 3 so that between the two 1's
there is one digit, between the two 2's there are two digits, and
between the two 3's there are three digits.
Use the clues to work out which cities Mohamed, Sheng, Tanya and
Bharat live in.
Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.
How many trains can you make which are the same length as Matt's, using rods that are identical?
Can you find all the different ways of lining up these Cuisenaire
Can you make a train the same length as Laura's but using three differently coloured rods? Is there only one way of doing it?
Find out what a "fault-free" rectangle is and try to make some of
Place the numbers 1 to 6 in the circles so that each number is the
difference between the two numbers just below it.
Can you work out how to balance this equaliser? You can put more
than one weight on a hook.
Katie had a pack of 20 cards numbered from 1 to 20. She arranged
the cards into 6 unequal piles where each pile added to the same
total. What was the total and how could this be done?
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
During the third hour after midnight the hands on a clock point in
the same direction (so one hand is over the top of the other). At
what time, to the nearest second, does this happen?
Can you fill in the empty boxes in the grid with the right shape
There are 78 prisoners in a square cell block of twelve cells. The
clever prison warder arranged them so there were 25 along each wall
of the prison block. How did he do it?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
You have two egg timers. One takes 4 minutes exactly to empty and
the other takes 7 minutes. What times in whole minutes can you
measure and how?