Fractals blog

Any thoughts, issues or solutions on the problem 'Any Win For Tennis'? Post them here.

Any thoughts or issues concerning the 10 Olympic Starters Problems? Discuss them here.

In this problem I look at a way of calculating $\sqrt{3}$.

This must be somewhere between $1$ and $2$. How do I know this? Now calculate the average of $\frac{3}{2}$ and 2 (which is 1.75) - this is a second approximation to $\sqrt 3$. i.e. we are saying that a better approximation to $\sqrt 3$ is $$x_{n+1} = \frac{(\frac{3}{x_n} + x_n)}{2}$$ where $x_n$ is an approximation to $\sqrt 3$ .

Consider a very big book, UltraRef, which refers in its index to every book which does not refer to itself in its index.

Does UltraRef refer to itself in its index? Be clear as to your explanation.

A mad robot sets off towards the North East on a journey from the origin. It travels in stages by moving forward and then rotating on the spot. It follows these pseudo-code instructions:

SUB JOURNEY

$\quad$DISTANCE = 1000
$\quad$WHILE (DISTANCE  > 0.001)
$\quad\quad$MOVE DISTANCE
$\quad\quad$STOP
$\quad\quad$ROTATE(90, DEGREES, CLOCKWISE)
$\quad\quad$DISTANCE = DISTANCE / 2
$\quad$END WHILE

$\quad$EXPLODE

END SUB

Where does the robot explode?
 

In July 1994 the Comet Shoemaker-Levy 9 struck the planet Jupiter at a speed of around 60km s$^{-1}$ . Although it broke up before impact, the core of the original comet was around 5 km diameter. Its estimated density was $0.3 - 0.7$ g cm$^{-3}$ . Imagine that such a comet had struck the earth. How much would it have changed the earth's velocity relative to the sun? (Note that the mass of the earth is about $6\times 10^{24}$ kg).

Choose any whole number $n$ and calculate $n^3 + 11n$.

Is the answer a multiple of 6 for all choices of $n$?

The challenge is to explain why or why not.