Prove that 3 times the sum of 3 squares is the sum of 4 squares. Rather easier, 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?

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 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.

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

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?

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...

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

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

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?

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

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

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

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

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 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.

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

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

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?

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.

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.

This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.

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

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

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

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

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.

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?

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

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.

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

If a sum invested gains 10% each year how long before it has doubled its value?

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. . . .

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

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.

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

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.

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

Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.

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

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.

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

How many winning lines can you make in a three-dimensional version of noughts and crosses?

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