Saturday, July 15, 2006

Elegance and Predictability

Good puzzles are true to a promise. They promise to offer a task that can be predicted. That is, they promise one solution that justifies itself better than any other. As any religious apologetic will agree, justification is not a solidified concept. And so the "real" solution to a puzzle can be evaluated aesthetically. Elegant is a nice way to describe a good puzzle and solution.

In my phonology classes our eternal struggle was to come up with the best set of rules or constraints to describe the process of sound alternation. Example: the received or underived pronunciation of water has a voiceless /t/ as the third morpheme - so does the received pronunciation of baton. In British English the /t/ remains unchanged all the way to its surface form. In American English it generally remains unchanged in baton, but in water the /t/ becomes a flap (we'll think of it as a /d/ although it technically isn't).

How can we describe this process in American English? In one common form of analysis (Sound Pattern of English, or SPE) we come up with formal rules describing phonological context and its effect on certain sounds. So we'll describe the context of /t/ in water:

Between two vowels after a stressed syllable.

Then we describe the context of /t/ in baton:

Between two vowels before a stressed syllable.

and looking at these two rules we suggest that a very elegant rule to account for the American pronunciation might be put into prose this way:

/t/ becomes /d/ when it occurs in an intervocalic post-stress position. That one simple rule accounts for the alternation in water and the unchanged form in baton.

For those who appreciate SPE this is an elegant solution because it accounts for the provided data and more besides. The word catnip doesn't meet both requirements: although the /t/ is post-stress it isn't intervocalic. The rules hold. More data will require some fine-tuning of the rules - but the point of the analysis remains - to account for as much data as possible by formulating as few rules as possible - rules that are as simple and natural as possible (so many possiblities...).

So puzzles...

There is a website out there called BrainBashers. If you browse through the site you'll find a few decent games and a few clever puzzles. But the site includes many puzzles that violate this maxim of elegance. I've included here two of their word-game puzzles.

First riddle:

This is an annoying paragraph, in which you try and work out what is unusual about it. Though, this paragraph has a quandary. A number of words have found a way to slink into this paragraph, to ruin your fun. What are those words? Do not try to run a utility to assist you, that would spoil all my attempts to absorb all of your skills in this mind blowing prank.

We've seen paragraphs like these and we know what to start looking for. Patterns - hidden messages (acrostics anagrams...) - oxymorons - double meanings - even the hard-to-catch missing letter - or the opposite of that: the sentence (like the one about the quick brown fox) that has every letter of the alphabet at least once.

Here's the banal solution:

"The paragraph has no words which contain a letter E, however, a few slipped in!"

So the solution is that there are only a few words with the letter E. This solution does not justify itself. There is no reason that this should be the most curious characteristic of the paragraph - is there? It's the most curious characteristic only because the author chose that as an ad hoc answer. It is not a characteristic that would be remarkable in any other paragraph or for any other puzzle. It is a solution created only for this puzzle.

Then we find this riddle.

"How many legs does an elephant have if you call its trunk a leg?"

This one is pretty predictable. It comes close to the old "pound of feathers vs. a pound of lead" riddle. But here we find that the author has chosen a meaning of the phrase "you call" that cannot be predicted and has been designated for this riddle only.

The given answer:
"Four: calling a trunk a leg does not make it a leg!"

They put the exclamation point on there. It doesn't deserve it.

If I'm given permission to call it a leg then it's a leg. It may not be a skeletal limb with phalanges metatarsals fibula tibia etc...but it's a leg. The author's claim that temporary labels are not to be confused with typical denotatum cannot be predictably applied to any other data.

One last example. On Jeopardy! last week one of the categories was Faux Fragrances. Here was the clue:

Alex: "So natural... so real... this salty, watery fluid secreted by sweat glands is finally here."
Sidney: "What are the armpits?" NO
John: "What is sweat?" NO

The answer: What is Perspiration.

I turned to my wife and began making the case that there were not enough guidelines established for "perspiration" to justify itself gracefully beyond armpits or sweat.

"Sit down" she said.


  1. "Do not try to run a utility to assist you, that would spoil all my attempts to absorb all of your skills in this mind blowing prank."

    Their prank manages to blow even a "utility's" mind, since the answer/rule is incomplete/broken. How could a program tell you most of the words don't have an e?

  2. Great post! Why can't people play by the rules of predictability?!?

  3. I refer you to which I linked to on our other blog a while ago. They make games that play on things like operant conditioning and people's assumptions of what constitutes a game.

    My favorites:

    How Much (The Subjectity Game)

    Please the Art Critic

    Click Between the Lights

  4. Here's a "riddle":

    This sentence has something curious about it that you can figure out if you think really hard.

    Answer: It has no "X" and only one "Y" and no "Z"s.

  5. Nonono Casey you missed it completely. The answer is

    - it has every vowel at least once.

    I was able to paint a romantic impressionist painting for the "please the art critic" game. Funny how it's pretty random but there's reason to think that there is some actual direction given.

    And I've not gotten past level 9 in the click game.


Thanks for reaching out.

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