Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

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.

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

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

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

Can you find a rule which connects consecutive triangular numbers?

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. . . .

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

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.

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

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?

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, 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?

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

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

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

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?

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

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

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}.

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.

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.

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

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

Can you make sense of these three proofs of Pythagoras' Theorem?

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.

Derive an equation which describes satellite dynamics.

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

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. . . .

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

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?

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

Relate these algebraic expressions to geometrical diagrams.

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

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

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.

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?

Fancy learning a bit more about rates of reaction, but don't know where to look? Come inside and find out more...

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

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.