Creating and manipulating expressions and formulae

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.

Label this plum tree graph to make it totally magic!

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.

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it works?

Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try to explain why this works.

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

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

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

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.