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.

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

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?

What can you say about the common difference of an AP where every term is prime?

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. . . .

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.

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.

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

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

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.

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)

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

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?

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.

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

Explore a number pattern which has the same symmetries in different bases.

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

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.

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

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.

Can you correctly order the steps in the proof of the formula for the sum of the first n terms in a geometric sequence?

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.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

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.

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

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?

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.

How many noughts are at the end of these giant numbers?

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

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

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

An inequality involving integrals of squares of functions.

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

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

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

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.

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

A introduction to how patterns can be deceiving, and what is and is not a proof.

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.