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

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?

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.

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

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. . . .

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

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

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

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

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?

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.

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

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 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.

Relate these algebraic expressions to geometrical diagrams.

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

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?

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 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.

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.

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

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.

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

Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T.

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

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...

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

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

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?

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

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

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

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?

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

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. . . .

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

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

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

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

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?

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

In a snooker game the brown ball was on the lip of the pocket but it could not be hit directly as the black ball was in the way. How could it be potted by playing the white ball off a cushion?

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

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =