Archive for November, 2008

Solution of the 12 balls problem

Posted by Ed on November 30, 2008

Last time I described the 12 balls problem. I also wrote about the 9 balls problem that is easier to solve.

Here is the solution to the 9 balls problem:

Here is the solution to the 12 balls problem:

Now, for the riddle with the 12 balls the trick is the following:
For the first weighing we put 1,2,3,4 on the left hand side and 5,6,7,8 on the right hand side. Suppose 1,2,3,4 goes down as in the picture below. What do you do?

The key step is to put two “heavy” balls on one side and one “light” ball on the other side, e.g. 1,2 on the left and 6 on the right as in the picture below. I leave it to you to find out what you can deduce if the left side goes down.

Besides, I created the pictures with Inkscape, a vector graphics program. I like it very much and it’s freeware.

Posted in mathematics | Tagged: , | 1 Comment »

LaTeX for WordPress

Posted by Ed on November 15, 2008

LaTeX is a typesetting system especially useful when a lot of formulas are used.

Testing LaTeX for WordPress:

1. SchrÃ¶dinger equation
$\left(- \frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x)\right) \psi(x,t) = i \hbar \frac{\partial}{\partial t}\psi(x,t)$
2. Some random formula
$\int_{0}^{1}\frac{x^{4}\left(1-x\right)^{4}}{1+x^{2}}dx=\frac{22}{7}-\pi$

If you’d like to use latex for WordPress read the the official description here.

Posted in computer science, mathematics, physics | Tagged: | 1 Comment »

The twelve balls problem

Posted by Ed on November 15, 2008

There are twelve balls that look equal. But one of them has a different weight, though you don’t know whether it is lighter or heavier. Your task is to find that different ball with a beam balance but you are only allowed to use the beam balance three times. How do you do it?

I finally solved the ‘twelve balls riddle’. It took me 3 months. Of course, as you can guess from the time this problem kept me busy, I found it quite hard. I learned something from it: if you really want to solve a problem sit down, take paper and pencil and work seriously on it. When I worked on the twelve balls riddle I often sat in the train trying to solve it in my head. But even when I had something to write on I was often too tired. Moreover, you can imagine it is not particularly comfortable writing down things in a train.
So what I mean with
seriously is that you should put all your effort into solving the problem. Create the best possible conditions that you can work under, e.g. by having a large and clean desktop. Work in a calm environment such as a library. And work with a clear mind and not when you are tired. Do not make it any harder than necessary by trying to imagine the solution solely in your head instead of using a notepad.
And guess what: when I finally found the solution, I was at home with paper and pencil.

If you like to solve the ‘twelve balls problem’ I recommend having a look at the ‘nine balls problem‘ first. It is much easier:
There are nine balls that look equal. But one of them is heavier. How do you find the heavier ball by using a beam balance only twice?
Note that this time you know that the different ball is heavier and that you can use the beam balance only twice.

I will post my solution to both problems later.

Posted in mathematics | Tagged: , | 2 Comments »

Posted by Ed on November 2, 2008

You can find some advice to the young mathematician in the Princeton Companion of Mathematics (PCM).

Besides, editor Timothy gowers calls the PCM as “Mathematics: A Very Long Introduction”, so to say the big brother of his other book, Mathematics: A very short introduction. I have already read the latter book and I enjoyed it very much.