This is an interactivity in which you have to sort into the correct
order the steps in the proof of the formula for the sum of a
Explore what happens when you draw graphs of quadratic equations
with coefficients based on a geometric sequence.
Here the diagram says it all. Can you find the diagram?
Generalise the sum of a GP by using derivatives to make the
coefficients into powers of the natural numbers.
Can you see how this picture illustrates the formula for the sum of
the first six cube numbers?
Three points A, B and C lie in this order on a line, and P is any
point in the plane. Use the Cosine Rule to prove the following
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0
what can you say about the triangle?
Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
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.
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 which gives an account of some properties of magic squares.
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.
Peter Zimmerman, a Year 13 student at Mill Hill County High School
in Barnet, London wrote this account of modulus arithmetic.
Tom writes about expressing numbers as the sums of three squares.
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.
Can you discover whether this is a fair game?
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.
In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
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.
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
Peter Zimmerman from Mill Hill County High School in Barnet, London
gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is
divisible by 33 for every non negative integer n.
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.
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?
We only need 7 numbers for modulus (or clock) arithmetic mod 7
including working with fractions. Explore how to divide numbers and
write fractions in modulus arithemtic.
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
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.
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
The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .
Prove that you cannot form a Magic W with a total of 12 or less or
with a with a total of 18 or more.
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. . . .
To find the integral of a polynomial, evaluate it at some special
points and add multiples of these values.
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 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!
Follow the hints and prove Pick's Theorem.
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .
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.
This is the second article on right-angled triangles whose edge lengths are whole numbers.
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
Take a complicated fraction with the product of five quartics top
and bottom and reduce this to a whole number. This is a numerical
example involving some clever algebra.
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.
What fractions can you divide the diagonal of a square into by simple folding?
Can you make sense of these three proofs of Pythagoras' Theorem?
Have a go at being mathematically negative, by negating these
This article stems from research on the teaching of proof and
offers guidance on how to move learners from focussing on
experimental arguments to mathematical arguments and deductive