How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
The nth term of a sequence is given by the formula n^3 + 11n . Find
the first four terms of the sequence given by this formula and the
first term of the sequence which is bigger than one million. . . .
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?
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Make a set of numbers that use all the digits from 1 to 9, once and
once only. Add them up. The result is divisible by 9. Add each of
the digits in the new number. What is their sum? Now try some. . . .
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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.
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
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
Find the smallest positive integer N such that N/2 is a perfect
cube, N/3 is a perfect fifth power and N/5 is a perfect seventh
Carry out cyclic permutations of nine digit numbers containing the
digits from 1 to 9 (until you get back to the first number). Prove
that whatever number you choose, they will add to the same total.
How many noughts are at the end of these giant numbers?
Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .
In the following sum the letters A, B, C, D, E and F stand for six
distinct digits. Find all the ways of replacing the letters with
digits so that the arithmetic is correct.
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number.
Cross out the numbers on the same row and column. Repeat this
process. Add up you four numbers. Why do they always add up to 34?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
What are the missing numbers in the pyramids?
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
Explore what happens when you draw graphs of quadratic equations
with coefficients based on a geometric sequence.
This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.
If you know the sizes of the angles marked with coloured dots in
this diagram which angles can you find by calculation?
What can you say about the angles on opposite vertices of any
cyclic quadrilateral? Working on the building blocks will give you
insights that may help you to explain what is special about them.
Take any prime number greater than 3 , square it and subtract one.
Working on the building blocks will help you to explain what is
special about your results.
This article stems from research on the teaching of proof and
offers guidance on how to move learners from focussing on
experimental arguments to mathematical arguments and deductive
L triominoes can fit together to make larger versions of
themselves. Is every size possible to make in this way?
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .
Four jewellers share their stock. Can you work out the relative values of their gems?
Can you see how this picture illustrates the formula for the sum of
the first six cube numbers?
If you think that mathematical proof is really clearcut and
universal then you should read this article.
Can you fit Ls together to make larger versions of themselves?
If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?
Is the mean of the squares of two numbers greater than, or less
than, the square of their means?
Use the numbers in the box below to make the base of a top-heavy
pyramid whose top number is 200.
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
I am exactly n times my daughter's age. In m years I shall be ... How old am I?
Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you
notice when successive terms are taken? What happens to the terms
if the fraction goes on indefinitely?
In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.
Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
Prove that, given any three parallel lines, an equilateral triangle
always exists with one vertex on each of the three lines.