In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?
I start with a red, a blue, a green and a yellow marble. I can
trade any of my marbles for three others, one of each colour. Can I
end up with exactly two marbles of each colour?
Baker, Cooper, Jones and Smith are four people whose occupations
are teacher, welder, mechanic and programmer, but not necessarily
in that order. What is each person’s occupation?
After some matches were played, most of the information in the
table containing the results of the games was accidentally deleted.
What was the score in each match played?
If you know the sizes of the angles marked with coloured dots in
this diagram which angles can you find by calculation?
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.
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable.
Decide which of these diagrams are traversable.
Here are some examples of 'cons', and see if you can figure out where the trick is.
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.
Here are three 'tricks' to amaze your friends. But the really
clever trick is explaining to them why these 'tricks' are maths not
magic. Like all good magicians, you should practice by trying. . . .
We have exactly 100 coins. There are five different values of
coins. We have decided to buy a piece of computer software for
39.75. We have the correct money, not a penny more, not a penny
less! Can. . . .
What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.
Find the area of the annulus in terms of the length of the chord
which is tangent to the inner circle.
Replace each letter with a digit to make this addition correct.
These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.
A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.
Write down a three-digit number Change the order of the digits to
get a different number Find the difference between the two three
digit numbers Follow the rest of the instructions then try. . . .
A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.
This jar used to hold perfumed oil. It contained enough oil to fill
granid silver bottles. Each bottle held enough to fill ozvik golden
goblets and each goblet held enough to fill vaswik crystal. . . .
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
Nine cross country runners compete in a team competition in which
there are three matches. If you were a judge how would you decide
who would win?
From a group of any 4 students in a class of 30, each has exchanged
Christmas cards with the other three. Show that some students have
exchanged cards with all the other students in the class. How. . . .
Four of these clues are needed to find the chosen number on this
grid and four are true but do nothing to help in finding the
number. Can you sort out the clues and find the number?
Six points are arranged in space so that no three are collinear.
How many line segments can be formed by joining the points in
Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?
Can you fit Ls together to make larger versions of themselves?
What are the missing numbers in the pyramids?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .
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.
Eight children enter the autumn cross-country race at school. How
many possible ways could they come in at first, second and third
A huge wheel is rolling past your window. What do you see?
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.
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?
Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.
This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
What happens when you add three numbers together? Will your answer be odd or even? How do you know?
This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.
Use your logical reasoning to work out how many cows and how many
sheep there are in each field.
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
Can you find all the 4-ball shuffles?
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?