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?
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?
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?
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?
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?
How many different symmetrical shapes can you make by shading triangles or squares?
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?
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?
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?
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
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 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.
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?
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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.
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?
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?
Two sudokus in one. Challenge yourself to make the necessary
Make your own double-sided magic square. But can you complete both
sides once you've made the pieces?
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.
Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
An extra constraint means this Sudoku requires you to think in
diagonals as well as horizontal and vertical lines and boxes of
Four numbers on an intersection that need to be placed in the
surrounding cells. That is all you need to know to solve this
A Sudoku with clues as ratios.
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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.
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?
A Sudoku with a twist.
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?
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
Rather than using the numbers 1-9, this sudoku uses the nine
different letters used to make the words "Advent Calendar".
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Can you recreate these designs? What are the basic units? What
movement is required between each unit? Some elegant use of
procedures will help - variables not essential.
Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...
Bellringers have a special way to write down the patterns they
ring. Learn about these patterns and draw some of your own.
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?
Make a pair of cubes that can be moved to show all the days of the
month from the 1st to the 31st.
A Sudoku that uses transformations as supporting clues.
Given the products of diagonally opposite cells - can you complete this Sudoku?
Investigate the smallest number of moves it takes to turn these
mats upside-down if you can only turn exactly three at a time.
A Sudoku with clues as ratios or fractions.
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 few extra challenges set by some young NRICH members.
Ben passed a third of his counters to Jack, Jack passed a quarter
of his counters to Emma and Emma passed a fifth of her counters to
Ben. After this they all had the same number of counters.
How many solutions can you find to this sum? Each of the different letters stands for a different number.