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 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.
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. . . .
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. . . .
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?
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.
Find the area of the annulus in terms of the length of the chord
which is tangent to the inner circle.
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.
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?
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. . . .
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.
The circumcentres of four triangles are joined to form a
quadrilateral. What do you notice about this quadrilateral as the
dynamic image changes? Can you prove your conjecture?
Prove that the shaded area of the semicircle is equal to the area of the inner circle.
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
If you know the sizes of the angles marked with coloured dots in
this diagram which angles can you find by calculation?
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
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. . . .
Prove that the internal angle bisectors of a triangle will never be
perpendicular to each other.
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.
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.
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.
Here are some examples of 'cons', and see if you can figure out where the trick is.
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.
Make a set of numbers that use all the digits from 1 to 9, once and
once only. Add them up. The result is divisible by 9. Add each of
the digits in the new number. What is their sum? Now try some. . . .
Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .
A quadrilateral inscribed in a unit circle has sides of lengths s1, s2, s3 and s4 where s1 ≤ s2 ≤ s3 ≤ s4.
Find a quadrilateral of this type for which s1= sqrt2 and show s1 cannot. . . .
What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.
Use the numbers in the box below to make the base of a top-heavy
pyramid whose top number is 200.
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. . . .
The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .
Carry out cyclic permutations of nine digit numbers containing the
digits from 1 to 9 (until you get back to the first number). Prove
that whatever number you choose, they will add to the same total.
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?
Can you find all the 4-ball shuffles?
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?
Is it possible to rearrange the numbers 1,2......12 around a clock
face in such a way that every two numbers in adjacent positions
differ by any of 3, 4 or 5 hours?
A huge wheel is rolling past your window. What do you see?
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?
Can you see how this picture illustrates the formula for the sum of
the first six cube 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?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
We are given a regular icosahedron having three red vertices. Show
that it has a vertex that has at least two red neighbours.
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.
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. . . .
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.
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.
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.
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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 how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
Can you discover whether this is a fair game?