An irregular tetrahedron is composed of four different triangles.
Can such a tetrahedron be constructed where the side lengths are 4,
5, 6, 7, 8 and 9 units of length?
How many models can you find which obey these rules?
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?
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?
You have been given nine weights, one of which is slightly heavier
than the rest. Can you work out which weight is heavier in just two
weighings of the balance?
Find out about Magic Squares in this article written for students. Why are they magic?!
Bellringers have a special way to write down the patterns they
ring. Learn about these patterns and draw some of your own.
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?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
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.
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
This challenge is to design different step arrangements, which must
go along a distance of 6 on the steps and must end up at 6 high.
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
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.
Investigate all the different squares you can make on this 5 by 5
grid by making your starting side go from the bottom left hand
point. Can you find out the areas of all these squares?
Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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.
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
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?
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?
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.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
In this investigation, you must try to make houses using cubes. If
the base must not spill over 4 squares and you have 7 cubes which
stand for 7 rooms, what different designs can you come up with?
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.
Penta people, the Pentominoes, always build their houses from five
square rooms. I wonder how many different Penta homes you can
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.
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
These two group activities use mathematical reasoning - one is
numerical, one geometric.
If you had 36 cubes, what different cuboids could you make?
Can you find all the ways to get 15 at the top of this triangle of numbers?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
This task follows on from Build it Up and takes the ideas into three dimensions!
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?
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?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?
When intergalactic Wag Worms are born they look just like a cube.
Each year they grow another cube in any direction. Find all the
shapes that five-year-old Wag Worms can be.
Find out what a "fault-free" rectangle is and try to make some of
Two sudokus in one. Challenge yourself to make the necessary
How many different triangles can you make on a circular pegboard that has nine pegs?
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.
How can you put five cereal packets together to make different
shapes if you must put them face-to-face?
Just four procedures were used to produce a design. How was it
done? Can you be systematic and elegant so that someone can follow