Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
Problem solving is at the heart of the NRICH site. All the problems
give learners opportunities to learn, develop or use mathematical
concepts and skills. Read here for more information.
Given any two polynomials in a single variable it is always
possible to eliminate the variable and obtain a formula showing the
relationship between the two polynomials. Try this one.
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. . . .
Explain why, when moving heavy objects on rollers, the object moves
twice as fast as the rollers. Try a similar experiment yourself.
Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
Generalise the sum of a GP by using derivatives to make the
coefficients into powers of the natural numbers.
Four jewellers share their stock. Can you work out the relative values of their gems?
Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and record your findings in truth tables.
An inequality involving integrals of squares of functions.
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.
Investigate circuits and record your findings in this simple introduction to truth tables and logic.
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.
Sort these mathematical propositions into a series of 8 correct
Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.
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.
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, B & C own a half, a third and a sixth of a coin collection.
Each grab some coins, return some, then share equally what they had
put back, finishing with their own share. How rich are they?
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...
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. . . .
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.
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.
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.
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.
An article which gives an account of some properties of magic 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.
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.
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?
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
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
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
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.
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.
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. . . .
A blue coin rolls round two yellow coins which touch. The coins are
the same size. How many revolutions does the blue coin make when it
rolls all the way round the yellow coins? Investigate for a. . . .
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.
In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
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.
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
Some diagrammatic 'proofs' of algebraic identities and