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.

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?

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

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.

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 (a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions)

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

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

Can you correctly order the steps in the proof of the formula for the sum of a geometric series?

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

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

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.

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.

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. . . .

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

Freddie Manners, of Packwood Haugh School in Shropshire solved an alphanumeric without using the extra information supplied and this article explains his reasoning.

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.

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?

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

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

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

An inequality involving integrals of squares of functions.

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.

Have a go at being mathematically negative, by negating these statements.

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

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

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

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.

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

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.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.

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.

Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .

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}.

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

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

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

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.

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

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.

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

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.

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