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 NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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 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?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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?
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2
litres. Find a way to pour 9 litres of drink from one jug to
another until you are left with exactly 3 litres in three of the
Using the statements, can you work out how many of each type of
rabbit there are in these pens?
This magic square has operations written in it, to make it into a
maze. Start wherever you like, go through every cell and go out a
total of 15!
Bellringers have a special way to write down the patterns they
ring. Learn about these patterns and draw some of your own.
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?
Find the smallest whole number which, when mutiplied by 7, gives a
product consisting entirely of ones.
A student in a maths class was trying to get some information from
her teacher. She was given some clues and then the teacher ended by
saying, "Well, how old are they?"
Countries from across the world competed in a sports tournament. Can you devise an efficient strategy to work out the order in which they finished?
Use the clues to work out which cities Mohamed, Sheng, Tanya and
Bharat live in.
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?
The clues for this Sudoku are the product of the numbers in adjacent squares.
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 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.
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?
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
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?
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?
There are 78 prisoners in a square cell block of twelve cells. The
clever prison warder arranged them so there were 25 along each wall
of the prison block. How did 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?
Can you find all the ways to get 15 at the top of this triangle of numbers?
This task follows on from Build it Up and takes the ideas into three dimensions!
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.
You have two egg timers. One takes 4 minutes exactly to empty and
the other takes 7 minutes. What times in whole minutes can you
measure and how?
An investigation that gives you the opportunity to make and justify
Given the products of adjacent cells, can you complete this Sudoku?
Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99
How many ways can you do it?
There are 44 people coming to a dinner party. There are 15 square
tables that seat 4 people. Find a way to seat the 44 people using
all 15 tables, with no empty places.
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...
Zumf makes spectacles for the residents of the planet Zargon, who
have either 3 eyes or 4 eyes. How many lenses will Zumf need to
make all the different orders for 9 families?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
This package contains a collection of problems from the NRICH
website that could be suitable for students who have a good
understanding of Factors and Multiples and who feel ready to take
on some. . . .
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
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?
Make a pair of cubes that can be moved to show all the days of the
month from the 1st to the 31st.
If these elves wear a different outfit every day for as many days
as possible, how many days can their fun last?