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.
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...
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
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?
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.
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?
Suppose A always beats B and B always beats C, then would you
expect A to beat C? Not always! What seems obvious is not always
true. Results always need to be proved in mathematics.
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.
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. . . .
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. . . .
Can you rearrange the cards to make a series of correct mathematical statements?
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?
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.
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. . . .
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
By considering powers of (1+x), show that the sum of the squares of
the binomial coefficients from 0 to n is 2nCn
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 any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
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. . . .
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
If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.
An inequality involving integrals of squares of functions.
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?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
L triominoes can fit together to make larger versions of
themselves. Is every size possible to make in this way?
Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...
An introduction to some beautiful results of Number Theory
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?
A point moves around inside a rectangle. What are the least and the
greatest values of the sum of the squares of the distances from the
To find the integral of a polynomial, evaluate it at some special
points and add multiples of these values.
If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?
A introduction to how patterns can be deceiving, and what is and is not a proof.
These proofs are wrong. Can you see why?
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.
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.
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. . . .
Can you invert the logic to prove these statements?
Prove that in every tetrahedron there is a vertex such that the
three edges meeting there have lengths which could be the sides of
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.
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. . . .
Can you work through these direct proofs, using our interactive
Follow the hints and prove Pick's Theorem.
Have a go at being mathematically negative, by negating these
Tom writes about expressing numbers as the sums of three squares.
An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?