Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are
natural numbers and 0 < a < b < c. Prove that there is
only one set of values which satisfy this equation.
Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?
There are some water lilies in a lake. The area that they cover doubles in size every day. After 17 days the whole lake is covered. How long did it take them to cover half the lake?
What is the last digit of the number 1 / 5^903 ?
115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
What are the last two digits of 2^(2^2003)?
Problem one was solved by 70% of the pupils. Problem 2 was solved
by 60% of them. Every pupil solved at least one of the problems.
Nine pupils solved both problems. How many pupils took the exam?
Find out about palindromic numbers by reading this article.
Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.
Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?
Imagine a machine with four coloured lights which respond to different rules. Can you find the smallest possible number which will make all four colours light up?
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
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?
Consider all two digit numbers (10, 11, . . . ,99). In writing down
all these numbers, which digits occur least often, and which occur
most often ? What about three digit numbers, four digit numbers. . . .
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. . . .
Can you find any perfect numbers? Read this article to find out more...
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.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.
This investigation is about happy numbers in the World of the Octopus where all numbers are written in base 8 ... Find all the fixed points and cycles for the happy number sequences in base 8.
A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.
The number 10112359550561797752808988764044943820224719 is called a
'slippy number' because, when the last digit 9 is moved to the
front, the new number produced is the slippy number multiplied by
Find the maximum value of 1/p + 1/q + 1/r where this sum is less than 1 and p, q, and r are positive integers.
A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
What is the smallest number with exactly 14 divisors?
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
Which numbers can we write as a sum of square numbers?
Take three whole numbers. The differences between them give you
three new numbers. Find the differences between the new numbers and
keep repeating this. What happens?
Work out how to light up the single light. What's the rule?
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?
Can you work out how many of each kind of pencil this student
In how many ways can a pound (value 100 pence) be changed into some combination of 1, 2, 5, 10, 20 and 50 pence coins?
Suppose you had to begin the never ending task of writing out the
natural numbers: 1, 2, 3, 4, 5.... and so on. What would be the
1000th digit you would write down.
When asked how old she was, the teacher replied: My age in years is
not prime but odd and when reversed and added to my age you have a
Find the five distinct digits N, R, I, C and H in the following
What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?
Visitors to Earth from the distant planet of Zub-Zorna were amazed
when they found out that when the digits in this multiplication
were reversed, the answer was the same! Find a way to explain. . . .
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?
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
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.
When the number x 1 x x x is multiplied by 417 this gives the
answer 9 x x x 0 5 7. Find the missing digits, each of which is
represented by an "x" .
This article explains how credit card numbers are defined and the check digit serves to verify their accuracy.
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
Robert noticed some interesting patterns when he highlighted square
numbers in a spreadsheet. Can you prove that the patterns will
Complete the following expressions so that each one gives a four
digit number as the product of two two digit numbers and uses the
digits 1 to 8 once and only once.
Using the digits 1, 2, 3, 4, 5, 6, 7 and 8, mulitply a two two digit numbers are multiplied to give a four digit number, so that the expression is correct. How many different solutions can you find?
Is it true that $99^n$ has 2n digits and $999^n$ has 3n digits? Investigate!