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?
Can you explain why a sequence of operations always gives you perfect squares?
Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?
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.
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
I am exactly n times my daughter's age. In m years I shall be ... How old am I?
Prove that, given any three parallel lines, an equilateral triangle
always exists with one vertex on each of the three lines.
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
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.
Investigate the number of points with integer coordinates on
circles with centres at the origin for which the square of the
radius is a power of 5.
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?
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
Three equilateral triangles ABC, AYX and XZB are drawn with the
point X a moveable point on AB. The points P, Q and R are the
centres of the three triangles. What can you say about triangle
This is an interactivity in which you have to sort the steps in the
completion of the square into the correct order to prove the
formula for the solutions of quadratic equations.
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
Here the diagram says it all. Can you find the diagram?
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.
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
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. . . .
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. . . .
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.
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.
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.
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.
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
Solve this famous unsolved problem and win a prize. Take a positive
integer N. If even, divide by 2; if odd, multiply by 3 and add 1.
Iterate. Prove that the sequence always goes to 4,2,1,4,2,1...
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.
An article which gives an account of some properties of magic squares.
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. . . .
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?
This is the second article on right-angled triangles whose edge lengths are whole numbers.
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.
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
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
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.
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
Follow the hints and prove Pick's Theorem.
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!
A polite number can be written as the sum of two or more
consecutive positive integers. Find the consecutive sums giving the
polite numbers 544 and 424. What characterizes impolite numbers?
If you think that mathematical proof is really clearcut and
universal then you should read this article.
L triominoes can fit together to make larger versions of
themselves. Is every size possible to make in this way?
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?
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 has. . . .
Which of these roads will satisfy a Munchkin builder?
Explore what happens when you draw graphs of quadratic equations
with coefficients based on a geometric sequence.
This problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .
Mark a point P inside a closed curve. Is it always possible to find
two points that lie on the curve, such that P is the mid point of
the line joining these two points?
Four identical right angled triangles are drawn on the sides of a
square. Two face out, two face in. Why do the four vertices marked
with dots lie on one line?