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?
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?
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?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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?
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.
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.
Find out about Magic Squares in this article written for students. Why are they magic?!
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?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
How many models can you find which obey these rules?
Bellringers have a special way to write down the patterns they
ring. Learn about these patterns and draw some of your own.
Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.
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?
The letters of the word ABACUS have been arranged in the shape of a
triangle. How many different ways can you find to read the word
ABACUS from this triangular pattern?
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.
A Sudoku with clues given as sums of entries.
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.
A Sudoku with clues as ratios.
Remember that you want someone following behind you to see where
you went. Can yo work out how these patterns were created and
Two sudokus in one. Challenge yourself to make the necessary
A Sudoku with a twist.
This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?
How many different symmetrical shapes can you make by shading triangles or squares?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
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 second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Solve this Sudoku puzzle whose clues are in the form of sums of the
numbers which should appear in diagonal opposite cells.
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
A Sudoku that uses transformations as supporting clues.
Four numbers on an intersection that need to be placed in the
surrounding cells. That is all you need to know to solve this
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?
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.
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
Find out what a "fault-free" rectangle is and try to make some of
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?
A Sudoku with clues as ratios or fractions.
Given the products of diagonally opposite cells - can you complete this Sudoku?
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 sudoku requires you to have "double vision" - two Sudoku's for
the price of one
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.