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?
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?"
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. . . .
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
Given the products of diagonally opposite cells - can you complete this Sudoku?
In a square in which the houses are evenly spaced, numbers 3 and 10
are opposite each other. What is the smallest and what is the
largest possible number of houses in the square?
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 that gives you the opportunity to make and justify
Find the smallest whole number which, when mutiplied by 7, gives a
product consisting entirely of ones.
Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.
Suppose we allow ourselves to use three numbers less than 10 and
multiply them together. How many different products can you find?
How do you know you've got them all?
Can you work out the arrangement of the digits in the square so
that the given products are correct? The numbers 1 - 9 may be used
once and once only.
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
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?
Find out about Magic Squares in this article written for students. Why are they magic?!
The planet of Vuvv has seven moons. Can you work out how long it is
between each super-eclipse?
Can you order the digits from 1-6 to make a number which is
divisible by 6 so when the last digit is removed it becomes a
5-figure number divisible by 5, and so on?
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?
How many different shaped boxes can you design for 36 sweets in one
layer? Can you arrange the sweets so that no sweets of the same
colour are next to each other in any direction?
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 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.
Place the numbers 1 to 8 in the circles so that no consecutive
numbers are joined by a line.
Place eight queens on an chessboard (an 8 by 8 grid) so that none
can capture any of the others.
On a digital clock showing 24 hour time, over a whole day, how many
times does a 5 appear? Is it the same number for a 12 hour clock
over a whole day?
Investigate the different ways you could split up these rooms so
that you have double the number.
A merchant brings four bars of gold to a jeweller. How can the
jeweller use the scales just twice to identify the lighter, fake
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!
Katie had a pack of 20 cards numbered from 1 to 20. She arranged
the cards into 6 unequal piles where each pile added to the same
total. What was the total and how could this be done?
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?
Stuart's watch loses two minutes every hour. Adam's watch gains one
minute every hour. Use the information to work out what time (the
real time) they arrived at the airport.
Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?
How could you put eight beanbags in the hoops so that there are
four in the blue hoop, five in the red and six in the yellow? Can
you find all the ways of doing this?
You cannot choose a selection of ice cream flavours that includes
totally what someone has already chosen. Have a go and find all the
different ways in which seven children can have ice cream.
This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.
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.
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.
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.
In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?
This task depends on groups working collaboratively, discussing and reasoning to agree a final product.
Can you find which shapes you need to put into the grid to make the
totals at the end of each row and the bottom of each column?
Find the product of the numbers on the routes from A to B. Which
route has the smallest product? Which the largest?
How many different symmetrical shapes can you make by shading triangles or squares?
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?
Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?
Two sudokus in one. Challenge yourself to make the necessary
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
Each clue number in this sudoku is the product of the two numbers in adjacent cells.
In the planet system of Octa the planets are arranged in the shape
of an octahedron. How many different routes could be taken to get
from Planet A to Planet Zargon?
What do you notice about the date 03.06.09? Or 08.01.09? This
challenge invites you to investigate some interesting dates
This challenge is to design different step arrangements, which must
go along a distance of 6 on the steps and must end up at 6 high.