Technology

This is a nice article in the NYT about a recently proposed solution to the famously unsolved mathematical question “Does P = NP?”

http://www.nytimes.com/2010/08/17/science/17proof.html

Essentially this question is about how long it takes to solve certain kinds of problems:  if a proposed solution to a problem can be checked in some reasonable amount of time, does that mean we can always solve the problem in a reasonable amount of time?  [Warning:  the definition of reasonable here may seem unreasonable.]

For example, it doesn’t require many operations to determine whether or not 7411 divides 748511;  even by hand, you can work it out in a few steps.  It requires significantly more operations, however, to find the prime factors of, say, 837751.  Essentially, P v NP asks “are problems that can be checked by computers (maybe lots and lots of computers working in parallel) necessarily solvable by computers?”  It is still an open question.

Another fascinating aspect of this particular open question is the role that the internet has played in bringing great mathematical minds together.  Proposed solutions can be instantly accessed and vetted by those capable of evaluating the arguments.  Such a community can work quickly and efficiently, not just to ascertain a proof’s validity, but to improve and refine it together.

This is a very cool (and somewhat technical) description of a simple way to use Mathematica to hide a secret message in an image file.

http://blog.wolfram.com/2010/07/08/doing-spy-stuff-with-mathematica/

The basic idea is that you erase the last digit (in binary) of each pixel’s “color channel”, and then use that spot to store part of the secret message.  Given an image, you can then recover the secret message by looking at that last digit in each channel.

This process does change the image somewhat, but not in a way that the normal human eye (with normal viewing equipment) would ever notice.  In the post, the author subtracts the adjusted image from the original:

To most viewers, the difference looks like this:

Only under extreme contrast can you actually see the real difference in the two.  Here, we see the invisible ink reappear!

All that’s left is to decode the colors and read your message.  But be careful–sending and receiving secret messages might get you some unwanted attention.