How many different journeys could you make if you were going to
visit four stations in this network? How about if there were five
stations? Can you predict the number of journeys for seven
Your challenge is to find the longest way through the network
following this rule. You can start and finish anywhere, and with
any shape, as long as you follow the correct order.
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?
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.
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?
A little mouse called Delia lives in a hole in the bottom of a
tree.....How many days will it be before Delia has to take the same
What happens when you add three numbers together? Will your answer be odd or even? How do you know?
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.
There are 44 people coming to a dinner party. There are 15 square
tables that seat 4 people. Find a way to seat the 44 people using
all 15 tables, with no empty places.
Make a pair of cubes that can be moved to show all the days of the
month from the 1st to the 31st.
Alice and Brian are snails who live on a wall and can only travel
along the cracks. Alice wants to go to see Brian. How far is the
shortest route along the cracks? Is there more than one way to go?
A Sudoku that uses transformations as supporting clues.
What do the digits in the number fifteen add up to? How many other
numbers have digits with the same total but no zeros?
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?
Lolla bought a balloon at the circus. She gave the clown six coins
to pay for it. What could Lolla have paid for the balloon?
A Sudoku with clues as ratios.
Given the products of diagonally opposite cells - can you complete
Ram divided 15 pennies among four small bags. He could then pay any sum of money from 1p to 15p without opening any bag. How many pennies did Ram put in each bag?
You need to find the values of the stars before you can apply normal Sudoku rules.
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.
Look carefully at the numbers. What do you notice? Can you make
another square using the numbers 1 to 16, that displays the same
Explore this how this program produces the sequences it does. What
are you controlling when you change the values of the variables?
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
Use your logical-thinking skills to deduce how much Dan's crisps
and ice-cream cost altogether.
A Sudoku with clues as ratios or fractions.
There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?
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?
Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possibilities that could come up?
Using the statements, can you work out how many of each type of
rabbit there are in these pens?
This problem is based on a code using two different prime numbers
less than 10. You'll need to multiply them together and shift the
alphabet forwards by the result. Can you decipher the code?
Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?
This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
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.
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.
Add the sum of the squares of four numbers between 10 and 20 to the
sum of the squares of three numbers less than 6 to make the square
of another, larger, number.
Can you make square numbers by adding two prime numbers together?
Ben has five coins in his pocket. How much money might he have?
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.
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Find the product of the numbers on the routes from A to B. Which
route has the smallest product? Which the largest?
This magic square has operations written in it, to make it into a
maze. Start wherever you like, go through every cell and go out a
total of 15!
Can you fill in this table square? The numbers 2 -12 were used to
generate it with just one number used twice.
How could you put eight beanbags in the hoops so that there are
four in the blue hoop, five in the red and six in the yellow? Can
you find all the ways of doing this?
Suppose we allow ourselves to use three numbers less than 10 and
multiply them together. How many different products can you find?
How do you know you've got them all?
In the planet system of Octa the planets are arranged in the shape
of an octahedron. How many different routes could be taken to get
from Planet A to Planet Zargon?
There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2
litres. Find a way to pour 9 litres of drink from one jug to
another until you are left with exactly 3 litres in three of the
Pentagram Pylons - can you elegantly recreate them? Or, the
European flag in LOGO - what poses the greater problem?
Here are four cubes joined together. How many other arrangements of
four cubes can you find? Can you draw them on dotty paper?
60 pieces and a challenge. What can you make and how many of the
pieces can you use creating skeleton polyhedra?