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

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

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

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...

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?

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?

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

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?

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

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

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

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.

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?

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.

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?

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?

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.

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.

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

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.

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

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

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

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

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.

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

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?

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. . . .

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

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

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

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

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

Can you find a rule which connects consecutive triangular numbers?

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

What is the total number of squares that can be made on a 5 by 5 geoboard?

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

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?