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.

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.

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.

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

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

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

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

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.

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?

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

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

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.

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

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

How good are you at finding the formula for a number pattern ?

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

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.

Can you explain what is going on in these puzzling number tricks?

Show that all pentagonal numbers are one third of a triangular number.

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2.

Can you find a rule which relates triangular numbers to square numbers?

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

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

Can you find a rule which connects consecutive triangular numbers?

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}.

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

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

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

If you know the perimeter of a right angled triangle, what can you say about the area?

There are unexpected discoveries to be made about square numbers...

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. . . .

Relate these algebraic expressions to geometrical diagrams.

What is special about the difference between squares of numbers adjacent to multiples of three?

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

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

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.

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

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. . . .

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?