Kipli's Cage

What do you mean?
How do you know?
What does that entail?
Simple Probability Problem

Fred wants to have a date this Saturday night. To increase his chances of getting one, Fred plans to ask both Amy and Beatrice. If only one accepts, then he is all set. If both accept then he'll flip a coin, and hope that the one who loses (or wins, depending on your perspective) hasn't seen Fatal Attraction.

Fred goes to the oracle at Delphi for some help. The oracle estimates the probability that Amy will accept at `.7`, the probability that Beatrice will accept at `.3`, and the probability that at least one rejects him at `.4`. Feeling confident about his odds, Fred heads off to find Amy and Beatrice. But along the way he runs into the blind Tiresias. Tiresias tells Fred that the probability that at least one woman will reject him is actually `.6`.

Who has given Fred the better odds, the Delphic oracle or Tiresias?

Solution:

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Spore: The Evolution Game

Sims creator Will Wright has announced that he is working on a new game---Spore: the game of evolution.

In the game, players start off as an amoeba in a 2D world, reminiscent of some early video games.

The aim is to grow and evolve generations of creatures, with players able to choose the physical attributes of their creation.

"You get to play every generation of the creature," said Mr Wright. "I want something boys can make scary things or casual gamers can make cute things."

He said the computer would analyse a creature's design and work out how it should behave.

In the demonstration, Mr Wright created a three-legged creature and said the software would work out how it should walk.

"This is the hardest piece of technology I have ever had to solve," he said.

I'm looking forward to seeing what they come up with. I enjoyed playing around with Creatures, but grew frustrated with my inability to get the little guys to eat. They would die of starvation (or illness) and I couldn't figure out what was wrong with them. But I loved the concept. Maybe I'll dig out my copy of Creatures 3 sometime and try again.

With the Spore game, I have to wonder: how with the Intelligent Design/Creationist folks deal with it? Will the game be banned from households because it teaches about that evilution idea? Will there be boycotts of stores that sell the game? Will its release be touted as one more example of how we have turned from God's ways, this time by trying to play God?

Or will it be championed by some as actually providing proof for an intelligent designer? If the logic of people like Michael Behe provide indication, I'll be willing to say 'yes'. After all, the game clearly requires the input of a player to determine the physical characteristics of the creature. I can see the argument now:

Q. What evidence is there for Intelligent Design?

A. Our investigations of the body, at the biochemical level, clearly show that we are composed of, literally, many tiny machines. Biologists even admit that they are machines! And have you ever seen a machine that wasn't designed by someone? No, of course not! Therefore, when we see machines such as we do in our cells, we must conclude that they had a designer.

But what's more, physicists tell us that we can view the universe as one big computer, each atom representing a bit of information. They have even estimated the computational power of the universe. But the computer game Spore, a game about evolution, requires a designer for input. So just as we need a designer to play Spore, we also need a designer to play the computer 'game' that the universe is carrying out all around us.

Actually, according to this 1UP article and this PC Games article, Spore will probably have relatively little to do with evolution as scientists know it. In any event, it looks like an interesting game.

Math B-Days

Happy Birthday Eugene Catalan!

Catalan is most famous for lending his name to the Catalan numbers, a number sequence that counts a wide variety of combinatorial objects. He was also expelled from the École Polytechnique in 1834, but I have not been able to track down why. According to the St. Andrew's history site, he was very active politically on the left-leaning side, so perhaps that had something to do with it.

Catalan also conjectured that the only integer solution to the equation `x^p - y^q = \pm 1` where all integers are greater than 1 is `x^p = 3^2`, `y^q = 2^3`. First conjectured in 1844, it was finally proved true by Preda Mihăilescu in 2002.

Here's an easier problem: Show that the only solution to the equation `2^p - 3^q = \pm 1` with `p,q > 1` is `p=3`, `q=2`.

Hint:

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Solution:

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Math on MTV

The MTV game show NEXT features a twist on the usual dating show:

We'll set you up on 5 dates. The minute you get annoyed, angry or just plain bored, simply kick 'em to the curb by saying "NEXT", and start over with someone new.

Often, the player who is making the decision to 'next' someone opts to continue through almost all of the contestants. But if he decides to stop on, say, the `i^(th)` person, what is the probability that he will have selected the one that he would rank highest of all contestants?

Rather than making decisions about dates, let's imagine that I am trying to find the best chinchilla from among $n$ chinchillas, if each is presented to me one at a time.

Some simplifying assumptions:

  • I can rank the chinchillas in terms of preference; there are no ties

  • I am allowed to see one chinchilla at a time, in random order, and must either decide to keep or reject him before moving on to the next; once a chinchilla is rejected, he cannot be reconsidered

  • I will not select a chinchilla that ranks lower than any chinchilla previously seen; I would prefer no chinchilla than one that I know is not my top pick (I'm shallow that way)

If I pick the `i^(th)` chinchilla I see, what is the probability that he is the top ranked chinchilla?

Solution:

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