Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
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.
Do you know how to find the area of a triangle? You can count the
squares. What happens if we turn the triangle on end? Press the
button and see. Try counting the number of units in the triangle
now. . . .
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
Can you discover whether this is a fair game?
Find the area of the annulus in terms of the length of the chord
which is tangent to the inner circle.
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. . . .
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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.
In how many ways can you arrange three dice side by side on a
surface so that the sum of the numbers on each of the four faces
(top, bottom, front and back) is equal?
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.
Some puzzles requiring no knowledge of knot theory, just a careful
inspection of the patterns. A glimpse of the classification of
knots and a little about prime knots, crossing numbers and. . . .
Make an eight by eight square, the layout is the same as a
chessboard. You can print out and use the square below. What is the
area of the square? Divide the square in the way shown by the red
dashed. . . .
A huge wheel is rolling past your window. What do you see?
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!
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
ABC is an equilateral triangle and P is a point in the interior of
the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP
must be less than 10 cm.
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.
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.
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.
Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?
Can you find all the 4-ball shuffles?
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. . . .
What happens when you add three numbers together? Will your answer be odd or even? How do you know?
A introduction to how patterns can be deceiving, and what is and is not a proof.
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
Take any whole number between 1 and 999, add the squares of the
digits to get a new number. Make some conjectures about what
happens in general.
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?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Can you fit Ls together to make larger versions of themselves?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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.
Use your logical reasoning to work out how many cows and how many
sheep there are in each field.
What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Here are some examples of 'cons', and see if you can figure out where the trick is.
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
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
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?
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.
In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.
Is it true that any convex hexagon will tessellate if it has a pair
of opposite sides that are equal, and three adjacent angles that
add up to 360 degrees?
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten.
Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
You have been given nine weights, one of which is slightly heavier
than the rest. Can you work out which weight is heavier in just two
weighings of the balance?
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.
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
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?