L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?
What is the largest number of intersection points that a triangle and a quadrilateral can have?
Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.
When is it impossible to make number sandwiches?
What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.
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.
Here are some examples of 'cons', and see if you can figure out where the trick is.
Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.
There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
Which set of numbers that add to 10 have the largest product?
Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?
Prove Pythagoras' Theorem using enlargements and scale factors.
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. . . .
Four jewellers share their stock. Can you work out the relative values of their gems?
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. . . .
Which hexagons tessellate?
What fractions can you divide the diagonal of a square into by simple folding?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .
Can you make sense of these three proofs of Pythagoras' Theorem?
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. . . .
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. . . .
Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?
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?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
This is the second article on right-angled triangles whose edge lengths are whole numbers.
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
Can you discover whether this is a fair game?
Kyle and his teacher disagree about his test score - who is right?
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.
Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.
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. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.
Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.
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 the areas of the trapezia in this sequence?
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?
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. . . .
Can you make sense of the three methods to work out the area of the kite in the square?
Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?
Can you use the diagram to prove the AM-GM inequality?
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
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?
The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?
Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?