If you know the sizes of the angles marked with coloured dots in
this diagram which angles can you find by calculation?
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. . . .
What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.
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?
Your partner chooses two beads and places them side by side behind a screen. What is the minimum number of guesses you would need to be sure of guessing the two beads and their positions?
Here are some examples of 'cons', and see if you can figure out where the trick is.
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.
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.
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. . . .
Can you find all the 4-ball shuffles?
Replace each letter with a digit to make this addition correct.
Eight children enter the autumn cross-country race at school. How
many possible ways could they come in at first, second and third
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.
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. . . .
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. . . .
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. . . .
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
Find the area of the annulus in terms of the length of the chord
which is tangent to the inner circle.
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?
Six points are arranged in space so that no three are collinear.
How many line segments can be formed by joining the points in
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?
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?
What are the missing numbers in the pyramids?
Can you fit Ls together to make larger versions of themselves?
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?
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.
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. . . .
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.
A introduction to how patterns can be deceiving, and what is and is not a proof.
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. . . .
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.
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.
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?
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.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
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.
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.
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Use your logical reasoning to work out how many cows and how many
sheep there are in each field.
What happens when you add three numbers together? Will your answer be odd or even? How do you know?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
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.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
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
Who said that adding couldn't be fun?