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.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
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.
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?
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.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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. . . .
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and record your findings in truth tables.
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?
Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .
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.
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
If you think that mathematical proof is really clearcut and universal then you should read this article.
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
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. . . .
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.
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. . . .
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
An article which gives an account of some properties of magic squares.
Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.
Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?
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 first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
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.
Investigate circuits and record your findings in this simple introduction to truth tables and logic.
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
Can you rearrange the cards to make a series of correct mathematical statements?
How many noughts are at the end of these giant numbers?
What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
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 iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.
Four jewellers share their stock. Can you work out the relative values of their gems?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Keep constructing triangles in the incircle of the previous triangle. What happens?
You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?
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.
Can you discover whether this is a fair game?
If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?
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.
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...
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
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 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!