Sort these mathematical propositions into a series of 8 correct statements.
Have a go at being mathematically negative, by negating these statements.
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?
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 reasoning.
These proofs are wrong. Can you see why?
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.
If you think that mathematical proof is really clearcut and universal then you should read this article.
Follow the hints and prove Pick's Theorem.
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.
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
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)
Can you invert the logic to prove these statements?
Can you rearrange the cards to make a series of correct mathematical statements?
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.
Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
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.
This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.
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?
Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?
If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
Starting with one of the mini-challenges, how many of the other mini-challenges will you invent for yourself?
An inequality involving integrals of squares of functions.
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.
The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
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?
Which of these roads will satisfy a Munchkin builder?
Explore a number pattern which has the same symmetries in different bases.
Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.
If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?
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.
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
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.
Here the diagram says it all. Can you find the diagram?
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.
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. . . .
Do you have enough information to work out the area of the shaded quadrilateral?
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
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.
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.
To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
An article which gives an account of some properties of magic squares.
Some diagrammatic 'proofs' of algebraic identities and inequalities.
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?