A combination mechanism for a safe comprises thirty-two tumblers
numbered from one to thirty-two in such a way that the numbers in
each wheel total 132... Could you open the safe?
What is the largest number you can make using the three digits 2, 3
and 4 in any way you like, using any operations you like? You can
only use each digit once.
There are exactly 3 ways to add 4 odd numbers to get 10. Find all
the ways of adding 8 odd numbers to get 20. To be sure of getting
all the solutions you will need to be systematic. What about. . . .
Investigate $1^n + 19^n + 20^n + 51^n + 57^n + 80^n + 82^n $ and $2^n + 12^n + 31^n + 40^n + 69^n + 71^n + 85^n$ for different values of n.
The sum of the first 'n' natural numbers is a 3 digit number in which all the digits are the same. How many numbers have been summed?
When I type a sequence of letters my calculator gives the product
of all the numbers in the corresponding memories. What numbers
should I store so that when I type 'ONE' it returns 1, and when I
type. . . .
How can we help students make sense of addition and subtraction of negative numbers?
Can you arrange the digits 1,2,3,4,5,6,7,8,9 into three 3-digit
numbers such that their total is close to 1500?
Find out about Magic Squares in this article written for students. Why are they magic?!
Whenever two chameleons of different colours meet they change
colour to the third colour. Describe the shortest sequence of
meetings in which all the chameleons change to green if you start
with 12. . . .
If each of these three shapes has a value, can you find the totals
of the combinations? Perhaps you can use the shapes to make the
You have four jugs of 9, 7, 4 and 2 litres capacity. The 9 litre
jug is full of wine, the others are empty. Can you divide the wine
into three equal quantities?
In this game the winner is the first to complete a row of three. Are some squares easier to land on than others?
Replace each letter with a digit to make this addition correct.
What is the sum of all the digits in all the integers from one to
Find a great variety of ways of asking questions which make 8.
Using 3 rods of integer lengths, none longer than 10 units and not
using any rod more than once, you can measure all the lengths in
whole units from 1 to 10 units. How many ways can you do this?
This challenge is to make up YOUR OWN alphanumeric. Each letter
represents a digit and where the same letter appears more than once
it must represent the same digit each time.
If you wrote all the possible four digit numbers made by using each
of the digits 2, 4, 5, 7 once, what would they add up to?
Find the numbers in this sum
This article suggests some ways of making sense of calculations involving positive and negative numbers.
Ann thought of 5 numbers and told Bob all the sums that could be made by adding the numbers in pairs. The list of sums is 6, 7, 8, 8, 9, 9, 10,10, 11, 12. Help Bob to find out which numbers Ann was. . . .
Fill in the numbers to make the sum of each row, column and
diagonal equal to 34. For an extra challenge try the huge American
Flag magic square.
Using some or all of the operations of addition, subtraction, multiplication and division and using the digits 3, 3, 8 and 8 each once and only once make an expression equal to 24.
Can you each work out the number on your card? What do you notice?
How could you sort the cards?
Bernard Bagnall recommends some primary school problems which use
numbers from the environment around us, from clocks to house
Investigate the different distances of these car journeys and find out how long they take.
Fill in the missing numbers so that adding each pair of corner
numbers gives you the number between them (in the box).
Skippy and Anna are locked in a room in a large castle. The key to that room, and all the other rooms, is a number. The numbers are locked away in a problem. Can you help them to get out?
This addition sum uses all ten digits 0, 1, 2...9 exactly once.
Find the sum and show that the one you give is the only
What happens when you add three numbers together? Will your answer be odd or even? How do you know?
Susie took cherries out of a bowl by following a certain pattern.
How many cherries had there been in the bowl to start with if she
was left with 14 single ones?
Find out what a Deca Tree is and then work out how many leaves
there will be after the woodcutter has cut off a trunk, a branch, a
twig and a leaf.
Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.
The Scot, John Napier, invented these strips about 400 years ago to
help calculate multiplication and division. Can you work out how to
use Napier's bones to find the answer to these multiplications?
Mrs Morgan, the class's teacher, pinned numbers onto the backs of
three children. Use the information to find out what the three
Can you draw a continuous line through 16 numbers on this grid so
that the total of the numbers you pass through is as high as
Start by putting one million (1 000 000) into the display of your
calculator. Can you reduce this to 7 using just the 7 key and add,
subtract, multiply, divide and equals as many times as you like?
Use your logical reasoning to work out how many cows and how many
sheep there are in each field.
In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice?
Can you substitute numbers for the letters in these sums?
Three dice are placed in a row. Find a way to turn each one so that
the three numbers on top of the dice total the same as the three
numbers on the front of the dice. Can you find all the ways to. . . .
Arrange three 1s, three 2s and three 3s in this square so that
every row, column and diagonal adds to the same total.
What do the digits in the number fifteen add up to? How many other
numbers have digits with the same total but no zeros?
Use 4 four times with simple operations so that you get the answer 12. Can you make 15, 16 and 17 too?
The clockmaker's wife cut up his birthday cake to look like a clock
face. Can you work out who received each piece?
This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.
Cassandra, David and Lachlan are brothers and sisters. They range
in age between 1 year and 14 years. Can you figure out their exact
ages from the clues?
Place the digits 1 to 9 into the circles so that each side of the
triangle adds to the same total.
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?