In this article, the NRICH team describe the process of selecting solutions for publication on the site.
This article for primary teachers suggests ways in which to help children become better at working systematically.
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.
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?
Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?
A package contains a set of resources designed to develop
students’ mathematical thinking. This package places a
particular emphasis on “being systematic” and is
designed to meet. . . .
How many different triangles can you make on a circular pegboard that has nine pegs?
Place the numbers 1 to 8 in the circles so that no consecutive
numbers are joined by a line.
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.
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?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
A challenging activity focusing on finding all possible ways of stacking rods.
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?
This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.
Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?
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.
This challenge extends the Plants investigation so now four or more children are involved.
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 help get a feel for this problem and to find out all the possible ways the balls could land.
Ben has five coins in his pocket. How much money might he have?
How many different triangles can you draw on the dotty grid which each have one dot in the middle?
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?
Can you work out how to balance this equaliser? You can put more
than one weight on a hook.
Make your own double-sided magic square. But can you complete both
sides once you've made the pieces?
Can you find all the different triangles on these peg boards, and
find their angles?
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 put the numbers 1 to 8 into the circles so that the four
calculations are correct?
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 put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?
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?
How many trains can you make which are the same length as Matt's, using rods that are identical?
How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?
Can you find all the different ways of lining up these Cuisenaire
Find out what a "fault-free" rectangle is and try to make some of
There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and
lollypops for 7p in the sweet shop. What could each of the children
buy with their money?
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?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Try out the lottery that is played in a far-away land. What is the
chance of winning?
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?
Can you cover the camel with these pieces?
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.
What is the smallest number of coins needed to make up 12 dollars and 83 cents?
Cut four triangles from a square as shown in the picture. How many
different shapes can you make by fitting the four triangles back
Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.
What happens when you try and fit the triomino pieces into these
In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?