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 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.

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

Can you prove that twice the sum of two squares always gives the sum of two squares?

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. . . .

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

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.

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?

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

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

Find the five distinct digits N, R, I, C and H in the following nomogram

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

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?

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

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

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.

Attach weights of 1, 2, 4, and 8 units to the four attachment points on the bar. Move the bar from side to side until you find a balance point. Is it possible to predict that position?

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

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

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

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.

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.

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.

To break down an algebraic fraction into partial fractions in which all the denominators are linear and all the numerators are constants you sometimes need complex numbers.

Can you hit the target functions using a set of input functions and a little calculus and algebra?

An algebra task which depends on members of the group noticing the needs of others and responding.

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

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

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?

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

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.

Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively.

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.

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

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

Relate these algebraic expressions to geometrical diagrams.

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

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

However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

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. . . .

Five equations... five unknowns... can you solve the system?

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

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?

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?