To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
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.
Find all the solutions to the this equation.
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.
Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
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.
Relate these algebraic expressions to geometrical diagrams.
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.
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.
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?
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
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 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. . . .
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
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. . . .
Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
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?
Four jewellers share their stock. Can you work out the relative values of their gems?
The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?
Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Can you explain why a sequence of operations always gives you perfect squares?
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 vertices?
Kyle and his teacher disagree about his test score - who is right?
I am exactly n times my daughter's age. In m years I shall be ... How old am I?
If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
Can you find the value of this function involving algebraic fractions for x=2000?
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.
A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.
By proving these particular identities, prove the existence of general cases.
Can you produce convincing arguments that a selection of statements about numbers are true?
An introduction to some beautiful results of Number Theory (a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions)
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?
Follow the hints and prove Pick's Theorem.
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.
Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?
This is the second article on right-angled triangles whose edge lengths are whole numbers.
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.
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. . . .
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
What is the largest number of intersection points that a triangle and a quadrilateral can have?
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. . . .
Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.
This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .