The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?
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?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
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?
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...
Which hexagons tessellate?
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.
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?
We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.
L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?
If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.
Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record. . . .
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.
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.
Some diagrammatic 'proofs' of algebraic identities and inequalities.
Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry
Can you discover whether this is a fair game?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .
A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.
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.
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.
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?
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.
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.
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
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.
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.
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.
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
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.
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
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.
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. . . .
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?
Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.
An article which gives an account of some properties of magic squares.
If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?
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 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.
The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.