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?
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. . . .
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. . . .
Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.
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?
Can you find any perfect numbers? Read this article to find out more...
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
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?
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.
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?
What is the last digit of the number 1 / 5^903 ?
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
What are the last two digits of 2^(2^2003)?
Find out about palindromic numbers by reading this article.
Robert noticed some interesting patterns when he highlighted square
numbers in a spreadsheet. Can you prove that the patterns will
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?
Which numbers can we write as a sum of square numbers?
Work out how to light up the single light. What's the rule?
Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
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?
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.
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.
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?
Find the five distinct digits N, R, I, C and H in the following
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?
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. . . .
Is there an efficient way to work out how many factors a large number has?
Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?
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.
What would you do if your teacher asked you add all the numbers from 1 to 100? Find out how Carl Gauss responded when he was asked to do just that.
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 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?
Can you work out how many of each kind of pencil this student
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.
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?
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.
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?
Is it true that $99^n$ has 2n digits and $999^n$ has 3n digits? Investigate!
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.
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?
How many positive integers less than or equal to 4000 can be
written down without using the digits 7, 8 or 9?
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.
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?
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
How many zeros are there at the end of the number which is the
product of first hundred positive integers?
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?