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?
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.
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.
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number.
Cross out the numbers on the same row and column. Repeat this
process. Add up you four numbers. Why do they always add up to 34?
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
A circle has centre O and angle POR = angle QOR. Construct tangents
at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q
lie inside, or on, or outside this circle?
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?
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. . . .
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.
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
Can you make sense of the three methods to work out the area of the kite in the square?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
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.
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.
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?
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. . . .
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. . . .
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?
What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
Eight children enter the autumn cross-country race at school. How
many possible ways could they come in at first, second and third
Replace each letter with a digit to make this addition correct.
Here are some examples of 'cons', and see if you can figure out where the trick is.
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.
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. . . .
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?
Prove that the shaded area of the semicircle is equal to the area of the inner 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 if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
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?
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.
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.
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!
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 lengths of the sides of a quadrilateral whose vertices are on a unit circle?
Keep constructing triangles in the incircle of the previous triangle. What happens?
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.
Which set of numbers that add to 10 have the largest product?
Prove Pythagoras' Theorem using enlargements and scale factors.
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.
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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.
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.
Find the area of the annulus in terms of the length of the chord
which is tangent to the inner circle.
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. . . .
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?