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

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

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

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

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

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. . . .

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.

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.

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}.

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

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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

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

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

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 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 inequality involving integrals of squares of functions.

An article about the strategy for playing The Triangle Game which appears on the NRICH site. It contains a simple lemma about labelling a grid of equilateral triangles within a triangular frame.

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?

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

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

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

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

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.

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

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.

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

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

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?

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

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.

Relate these algebraic expressions to geometrical diagrams.

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

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)

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

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.

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?

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.

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.

When is it impossible to make number sandwiches?

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

Can you make sense of these three proofs of Pythagoras' Theorem?

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.

What fractions can you divide the diagonal of a square into by simple folding?