Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

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. . . .

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

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?

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 any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

Starting with one of the mini-challenges, how many of the other mini-challenges will you invent for yourself?

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 serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

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.

Can you work out where the blue-and-red brick roads end?

By proving these particular identities, prove the existence of general cases.

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

Relate these algebraic expressions to geometrical diagrams.

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

Can you rearrange the cards to make a series of correct mathematical statements?

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 statement.

Sort these mathematical propositions into a series of 8 correct statements.

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

If you think that mathematical proof is really clearcut and universal then you should read this article.

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?

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

What is the largest number of intersection points that a triangle and a quadrilateral can have?

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

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.

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.

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.

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?

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

Can you find the value of this function involving algebraic fractions for x=2000?

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

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.

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.

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

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?

Can you work through these direct proofs, using our interactive proof sorters?