How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
Can you shunt the trucks so that the Cattle truck and the Sheep
truck change places and the Engine is back on the main line?
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?
Take a rectangle of paper and fold it in half, and half again, to
make four smaller rectangles. How many different ways can you fold
What is the best way to shunt these carriages so that each train
can continue its journey?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
A tetromino is made up of four squares joined edge to edge. Can
this tetromino, together with 15 copies of itself, be used to cover
an eight by eight chessboard?
10 space travellers are waiting to board their spaceships. There
are two rows of seats in the waiting room. Using the rules, where
are they all sitting? Can you find all the possible ways?
A challenging activity focusing on finding all possible ways of stacking rods.
Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?
You have 4 red and 5 blue counters. How many ways can they be
placed on a 3 by 3 grid so that all the rows columns and diagonals
have an even number of red counters?
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
Using different numbers of sticks, how many different triangles are
you able to make? Can you make any rules about the numbers of
sticks that make the most triangles?
Investigate the smallest number of moves it takes to turn these
mats upside-down if you can only turn exactly three at a time.
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....?
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?
In how many ways can you fit two of these yellow triangles
together? Can you predict the number of ways two blue triangles can
be fitted together?
Can you put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?
In how many ways can you stack these rods, following the rules?
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
Make your own double-sided magic square. But can you complete both
sides once you've made the pieces?
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?
How many different symmetrical shapes can you make by shading triangles or squares?
Take 5 cubes of one colour and 2 of another colour. How many
different ways can you join them if the 5 must touch the table and
the 2 must not touch the table?
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?
These practical challenges are all about making a 'tray' and covering it with paper.
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.
Can you find all the different ways of lining up these Cuisenaire
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
How many different triangles can you make on a circular pegboard that has nine pegs?
An activity making various patterns with 2 x 1 rectangular tiles.
A magician took a suit of thirteen cards and held them in his hand
face down. Every card he revealed had the same value as the one he
had just finished spelling. How did this work?
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?
Cut four triangles from a square as shown in the picture. How many
different shapes can you make by fitting the four triangles back
This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.
Four numbers on an intersection that need to be placed in the
surrounding cells. That is all you need to know to solve this
A few extra challenges set by some young NRICH members.
How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?
Bellringers have a special way to write down the patterns they
ring. Learn about these patterns and draw some of your own.
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
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?
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?
Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.
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 take a three by three square on a 1-10 addition square and
multiply the diagonally opposite numbers together, what is the
difference between these products. Why?
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?
Two sudokus in one. Challenge yourself to make the necessary
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 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?