An inequality involving integrals of squares of functions.
Sort these mathematical propositions into a series of 8 correct statements.
Follow the hints and prove Pick's Theorem.
By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
An introduction to some beautiful results of Number Theory
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 frame.
Can you work through these direct proofs, using our interactive proof sorters?
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...
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.
Here the diagram says it all. Can you find the diagram?
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.
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
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?
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.
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
Can you invert the logic to prove these 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?
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
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. . . .
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.
To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
Can you work out where the blue-and-red brick roads end?
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?
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.
Tom writes about expressing numbers as the sums of three squares.
These proofs are wrong. Can you see why?
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. . . .
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.
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?
Have a go at being mathematically negative, by negating these statements.
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.
Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.
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.
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.
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.
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!
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.
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
What can you say about the common difference of an AP where every term is prime?
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. . . .
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.
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
This is the second article on right-angled triangles whose edge lengths are whole numbers.
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's 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.
This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.
Can the pdfs and cdfs of an exponential distribution intersect?