What is the Hardest Word to Guess in Hangman?

hangmanIn 2007 Jon McLoone used Mathematica to create a Hangman game pitting the computer guesser against the human word-selector.  As his daughter became old enough to play against the demonstration, and old enough to get frustrated with the computer guesser always winning, she asked her dad the obvious question:  to beat the computer, what are the best words to choose?

Surprised that he had not considered such a good question himself, McLoone set about playing 15 million games of Hangman (automated, I imagine) using every word in the dictionary and arming the computer with a number of different letter-guessing-strategies.  The word that the computer failed to guess the most often was somewhat surprising.

So what kinds of strategies make the best guesser?  And to counter that, what kinds of strategies should the word-selector employ?

P v NP and Collaborative Mathematics

thinking computerThis 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.

Events with Probability Zero

Shirt

I just donated this shirt to the Salvation Army.  I estimate the probability that I will ever see this shirt again to be zero.  In fact, forget estimating:  I think the probability that I will see this shirt again is exactly zero.

In fact, I’ll go even further than that:  I claim that the probability that anyone who reads this ever sees this shirt is exactly zero.  Not .0000000000001.  Zero.

Even if there are infinitely many possible scenarios in which this shirt is seen by someone, this number is dwarfed by the infinitude of possible scenarios in total.  Thus

moderate infinity  ÷  really big infinity = zero

I dropped it off at the Salvation Army in Brooklyn.  Let me know if you see it:  evidence will be required to claim your prize.

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