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

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

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

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. . . .

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

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

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.

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 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.

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.

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

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)

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

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?

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

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

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

An inequality involving integrals of squares of functions.

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.

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

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.

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.

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

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.

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

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

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?

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.

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.

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

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

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}.

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

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

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.

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

When is it impossible to make number sandwiches?

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

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

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

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

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

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.

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.

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.