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. . . .
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.
This is the second article on right-angled triangles whose edge lengths are whole numbers.
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!
Prove Pythagoras' Theorem using enlargements and scale factors.
I am exactly n times my daughter's age. In m years I shall be ... How old am I?
Prove that the shaded area of the semicircle is equal to the area of 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. . . .
Find the area of the annulus in terms of the length of the chord
which is tangent to the inner circle.
Prove that, given any three parallel lines, an equilateral triangle
always exists with one vertex on each of the three lines.
It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?
Four identical right angled triangles are drawn on the sides of a
square. Two face out, two face in. Why do the four vertices marked
with dots lie on one line?
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?
Find the missing angle between the two secants to the circle when
the two angles at the centre subtended by the arcs created by the
intersections of the secants and the circle are 50 and 120 degrees.
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.
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
A blue coin rolls round two yellow coins which touch. The coins are
the same size. How many revolutions does the blue coin make when it
rolls all the way round the yellow coins? Investigate for a. . . .
Can you make sense of the three methods to work out the area of the kite in the square?
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. . . .
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.
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?
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.
Can you make sense of these three proofs of Pythagoras' Theorem?
What fractions can you divide the diagonal of a square into by simple folding?
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Explore what happens when you draw graphs of quadratic equations
with coefficients based on a geometric sequence.
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.
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.
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.
If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?
An article which gives an account of some properties of magic squares.
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 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.
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
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.
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Use the numbers in the box below to make the base of a top-heavy
pyramid whose top number is 200.
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?
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?
Can you fit Ls together to make larger versions of themselves?
Find the smallest positive integer N such that N/2 is a perfect
cube, N/3 is a perfect fifth power and N/5 is a perfect seventh
If you know the sizes of the angles marked with coloured dots in
this diagram which angles can you find by calculation?
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. . . .
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. . . .
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.