The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.
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.
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?
I am exactly n times my daughter's age. In m years I shall be exactly (n-1) times her age. In m2 years I shall be exactly (n-2) times her age. After that I shall never again be an exact multiple of. . . .
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
An article about the strategy for playing The Triangle Game which
appears on the NRICH site. It contains a simple lemma about
labelling a grid of equilateral triangles within a triangular
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
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?
Find all positive integers a and b for which the two equations:
x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.
Can you rearrange the cards to make a series of correct
Can you invert the logic to prove these statements?
An article which gives an account of some properties of magic squares.
Can you work through these direct proofs, using our interactive
Sort these mathematical propositions into a series of 8 correct
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
Peter Zimmerman, a Year 13 student at Mill Hill County High School
in Barnet, London wrote this account of modulus arithmetic.
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.
Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
Prove that, given any three parallel lines, an equilateral triangle
always exists with one vertex on each of the three lines.
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.
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.
If you think that mathematical proof is really clearcut and
universal then you should read this article.
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.
Here the diagram says it all. Can you find the diagram?
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. . . .
With n people anywhere in a field each shoots a water pistol at the
nearest person. In general who gets wet? What difference does it
make if n is odd or even?
The sum of any two of the numbers 2, 34 and 47 is a perfect square.
Choose three square numbers and find sets of three integers with
this property. Generalise to four integers.
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. . . .
Take a number, add its digits then multiply the digits together,
then multiply these two results. If you get the same number it is
an SP number.
Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.
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?
Prove that in every tetrahedron there is a vertex such that the
three edges meeting there have lengths which could be the sides of
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 connected graph is a graph in which we can get from any vertex to
any other by travelling along the edges. A tree is a connected
graph with no closed circuits (or loops. Prove that every tree. . . .
Which of these roads will satisfy a Munchkin builder?
Clearly if a, b and c are the lengths of the sides of a triangle and the triangle is equilateral then
a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true, and if so can you prove it? That is if. . . .
Take any whole number q. Calculate q^2 - 1. Factorize
q^2-1 to give two factors a and b (not necessarily q+1 and q-1). Put c = a + b + 2q . Then you will find that ab+1 , bc+1 and ca+1 are all. . . .
These proofs are wrong. Can you see why?
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.
Can you work out where the blue-and-red brick roads end?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?
Euler found four whole numbers such that the sum of any two of the
numbers is a perfect square. Three of the numbers that he found are
a = 18530, b=65570, c=45986. Find the fourth number, x. You. . . .
L triominoes can fit together to make larger versions of
themselves. Is every size possible to make in this way?
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
An introduction to some beautiful results of Number Theory
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?