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?
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?
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?
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...
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?
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 NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Charlie and Lynne 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
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?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
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?
How many different symmetrical shapes can you make by shading triangles or squares?
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 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?
If you are given the mean, median and mode of five positive whole
numbers, can you find the numbers?
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.
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?
Bellringers have a special way to write down the patterns they
ring. Learn about these patterns and draw some of your own.
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.
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 replace the letters with numbers? Is there only one
solution in each case?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Complete the magic square using the numbers 1 to 25 once each. Each
row, column and diagonal adds up to 65.
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?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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?
Use the differences to find the solution to this Sudoku.
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
Can you use this information to work out Charlie's house number?
Move your counters through this snake of cards and see how far you
can go. Are you surprised by where you end up?
Find out about Magic Squares in this article written for students. Why are they magic?!
An extra constraint means this Sudoku requires you to think in
diagonals as well as horizontal and vertical lines and boxes of
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
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.
Can you make square numbers by adding two prime numbers together?
An investigation that gives you the opportunity to make and justify
A few extra challenges set by some young NRICH members.
Find out what a "fault-free" rectangle is and try to make some of
This Sudoku, based on differences. Using the one clue number can you find the solution?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Add the sum of the squares of four numbers between 10 and 20 to the
sum of the squares of three numbers less than 6 to make the square
of another, larger, number.
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.
The discs for this game are kept in a flat square box with a square
hole for each disc. Use the information to find out how many discs
of each colour there are in the box.
Find the smallest whole number which, when mutiplied by 7, gives a
product consisting entirely of ones.
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.
If you take a three by three square on a 1-10 addition square and
multiply the diagonally opposite numbers together, what is the
difference between these products. Why?