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?
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.
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?
Find out what a "fault-free" rectangle is and try to make some of
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?
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?
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 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 symmetrical shapes can you make by shading triangles or squares?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Use the clues to work out which cities Mohamed, Sheng, Tanya and
Bharat live in.
Try out the lottery that is played in a far-away land. What is the
chance of winning?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
Two sudokus in one. Challenge yourself to make the necessary
A Sudoku with a twist.
This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.
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?
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?
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?
A Sudoku with clues as ratios.
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?
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?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
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 arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?
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?
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?
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.
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....?
Find a cuboid (with edges of integer values) that has a surface
area of exactly 100 square units. Is there more than one? Can you
find them all?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Solve this Sudoku puzzle whose clues are in the form of sums of the
numbers which should appear in diagonal opposite cells.
Let's suppose that you are going to have a magazine which has 16
pages of A5 size. Can you find some different ways to make these
pages? Investigate the pattern for each if you number the pages.
My two digit number is special because adding the sum of its digits
to the product of its digits gives me my original number. What
could my number be?
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?
In this matching game, you have to decide how long different events take.
A Sudoku that uses transformations as supporting clues.
How many different triangles can you make on a circular pegboard that has nine pegs?
Arrange the four number cards on the grid, according to the rules,
to make a diagonal, vertical or horizontal line.
Can you find all the different ways of lining up these Cuisenaire
Investigate the smallest number of moves it takes to turn these
mats upside-down if you can only turn exactly three at a time.