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...
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.
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?
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
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
An inequality involving integrals of squares of functions.
Here the diagram says it all. Can you find the diagram?
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
By considering powers of (1+x), show that the sum of the squares of
the binomial coefficients from 0 to n is 2nCn
Can you rearrange the cards to make a series of correct mathematical statements?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
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 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 serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
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?
In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
I am exactly n times my daughter's age. In m years I shall be ... How old am I?
Some diagrammatic 'proofs' of algebraic identities and
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.
Keep constructing triangles in the incircle of the previous triangle. What happens?
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.
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. . . .
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. . . .
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. . . .
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. . . .
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. . . .
Prove that, given any three parallel lines, an equilateral triangle
always exists with one vertex on each of the three lines.
To find the integral of a polynomial, evaluate it at some special
points and add multiples of these values.
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?
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...
Can you work through these direct proofs, using our interactive
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?
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?
Can you work out where the blue-and-red brick roads end?
Follow the hints and prove Pick's Theorem.
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
A introduction to how patterns can be deceiving, and what is and is not a proof.
These proofs are wrong. Can you see why?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
Prove that in every tetrahedron there is a vertex such that the
three edges meeting there have lengths which could be the sides of
Can you explain why a sequence of operations always gives you perfect squares?
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.
An introduction to some beautiful results of Number Theory
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. . . .
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.
There are 12 identical looking coins, one of which is a fake. The
counterfeit coin is of a different weight to the rest. What is the
minimum number of weighings needed to locate the fake coin?