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?
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....?
If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?
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 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?
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
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?
Can you work out how to balance this equaliser? You can put more
than one weight on a hook.
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?
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.
Can you cover the camel with these pieces?
What happens when you try and fit the triomino pieces into these
An activity making various patterns with 2 x 1 rectangular tiles.
Use the clues to colour each square.
This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .
Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.
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?
These practical challenges are all about making a 'tray' and covering it with paper.
What happens when you add three numbers together? Will your answer be odd or even? How do you know?
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
What is the best way to shunt these carriages so that each train
can continue its journey?
How many different rhythms can you make by putting two drums on the
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
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.
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
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?
This problem focuses on Dienes' Logiblocs. What is the same and
what is different about these pairs of shapes? Can you describe the
shapes in the picture?
You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
In this town, houses are built with one room for each person. There
are some families of seven people living in the town. In how many
different ways can they build their houses?
Let's say you can only use two different lengths - 2 units and 4
units. Using just these 2 lengths as the edges how many different
cuboids can you make?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
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?
Find your way through the grid starting at 2 and following these
operations. What number do you end on?
How many different triangles can you draw on the dotty grid which each have one dot in the middle?
The planet of Vuvv has seven moons. Can you work out how long it is
between each super-eclipse?
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.
Can you put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?
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?
Place eight queens on an chessboard (an 8 by 8 grid) so that none
can capture any of the others.
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.
Place eight dots on this diagram, so that there are only two dots
on each straight line and only two dots on each circle.
Investigate the different ways you could split up these rooms so
that you have double the number.
How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Place the numbers 1 to 6 in the circles so that each number is the
difference between the two numbers just below it.