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?

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

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.

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

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

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

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

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

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

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

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

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?

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

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?

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

Solve the system of equations: ab = 1 bc = 2 cd = 3 de = 4 ea = 6

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?

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.

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.

Several graphs of the sort occurring commonly in biology are given. How many processes can you map to each graph?

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 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?

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

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

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}.

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.

Relate these algebraic expressions to geometrical diagrams.

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.

Two cyclists, practising on a track, pass each other at the starting line and go at constant speeds... Can you find lap times that are such that the cyclists will meet exactly half way round the. . . .

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.

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.

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

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.

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

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.

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

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 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 functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?

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

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

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.