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Route to Root

A sequence of numbers x1, x2, x3, ... starts with x1 = 2, and, if you know any term xn, you can find the next term xn+1 using the formula: xn+1 = (xn + 3/xn)/2 . Calculate the first six terms of this sequence. What do you notice? Calculate a few more terms and find the squares of the terms. Can you prove that the special property you notice about this sequence will apply to all the later terms of the sequence? Write down a formula to give an approximation to the cube root of a number and test it for the cube root of 3 and the cube root of 8. How many terms of the sequence do you have to take before you get the cube root of 8 correct to as many decimal places as your calculator will give? What happens when you try this method for fourth roots or fifth roots etc.?

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Divided Differences

When in 1821 Charles Babbage invented the `Difference Engine' it was intended to take over the work of making mathematical tables by the techniques described in this article.

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Tournament Scheduling

Scheduling games is a little more challenging than one might desire. Here are some tournament formats that sport schedulers use.

Up a Semitone?

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Have you managed to solve the entire Stage 5 Cipher Challenge? Solutions are now closed, but perhaps you want to take up the full challenge.


Successful solvers of this part were

Patrick from Woodbridge School, England
An Anonymous Solver from Somewhere in the US
Joseph from Hong Kong

 

The solution was:

The keyword for the last cipher was pi. The technique used in this cipher is a Caesar square. It's probably a bit obvious having a square number of characters in the message. The next one might well be a combination of the techniques used so far.