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.

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

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.

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

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

Which of these triangular jigsaws are impossible to finish?

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.

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.

An inequality involving integrals of squares of functions.

Kyle and his teacher disagree about his test score - who is right?

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

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

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

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.

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

When is it impossible to make number sandwiches?

Can you explain why a sequence of operations always gives you perfect squares?

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

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.

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?

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.

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

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.

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?

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

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?

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

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.

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.

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

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

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

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!

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

This is the second article on right-angled triangles whose edge lengths are whole numbers.

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.

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

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?

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

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

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

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

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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