A introduction to how patterns can be deceiving, and what is and is not a proof.
Powers of numbers behave in surprising ways. Take a look at some of
these and try to explain why they are true.
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.
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. . . .
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. . . .
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?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
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.
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.
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?
Investigate the sequences obtained by starting with any positive 2
digit number (10a+b) and repeatedly using the rule 10a+b maps to
10b-a to get the next number in the sequence.
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?
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?
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.
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. . . .
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
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. . . .
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?
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.
Prove that the shaded area of the semicircle is equal to the area of the inner circle.
Prove that the internal angle bisectors of a triangle will never be
perpendicular to each other.
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
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 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?
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.
A composite number is one that is neither prime nor 1. Show that
10201 is composite in any base.
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
Can you discover whether this is a fair game?
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.
Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
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.
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. . . .
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) =
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
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?
If you know the sizes of the angles marked with coloured dots in
this diagram which angles can you find by calculation?
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 three prime numbers, all greater than 3, form an
arithmetic progression then the common difference is divisible by
6. What if one of the terms is 3?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
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.
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!
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 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.
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. . . .
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 article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
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.