The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.
Prove Pythagoras' Theorem using enlargements and scale factors.
An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
An article which gives an account of some properties of magic squares.
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. . . .
This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
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.
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?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
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.
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. . . .
The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?
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 convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
Which set of numbers that add to 10 have the largest product?
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?
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.
The sums of the squares of three related numbers is also a perfect square - can you explain why?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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. . . .
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?
L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
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?
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?
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?
When is it impossible to make number sandwiches?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .
What fractions can you divide the diagonal of a square into by simple folding?
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
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.
Can you rearrange the cards to make a series of correct mathematical statements?
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. . . .
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 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 how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
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.
A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.
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?