Sunday, May 18, 2008

Randomness and Reason

This past term, it fell to me to teach the perennial service course in statistics at Bennington College. In addition to covering various standard topics (descriptive statistics, significance testing, correlations, etc.), I spent a great deal of time with my beginning students on the theme of probabilistic reasoning based on evidence, as codified by Bayes's Theorem. I needed to do this if we were going to enter the richly confusing space between randomness and reason appropriately armed. I have assembled some of the most informal of the course lectures here. By way of apology for the overall state of things, I confess that during a busy term I struck off some of these mini-essays during the wee hours of the morning before class. (I thank my students for giving me a lot of room to experiment and improvise during the semester.) The lectures are collected with the vague idea that I might fix them up at some point in the future. Any comments are therefore very welcome.

Saturday, May 17, 2008

Four-Letter Words

The Change-a-Word is a staple of supermarket puzzle books. In Change-a-Words, you have to change a word like GULF into a word like YAWN by changing one letter at a time. (So you might say: GULF, GULL, BULL, BURL, BURN, BARN, YARN, YAWN.)

These puzzles are sort of fun, but they are mostly amusing for the variations they inspire (see e.g. my 75th Anniversary Lecture), and for the logical and linguistic questions they raise. For example,

1. Can most words be reached from most words?

2. Which words are "furthest" from each other? Specifically, if the shortest chain of words connecting A and B has length N, then which words A and B maximize N, and what is this maximal value of N?

I consulted my handy electronic dictionary to answer these questions.

As to question 1: For four-letter words, it appears to be the case that about 95% of the words are connected in a massive "supercontinent." This means most four-letter words can indeed be reached from most four-letter words - hence the steady stream of supermarket Change-a-Word books.

The 5% of four-letter-words lying "offshore" are either isolated "island words" (can you think of any?), or isolated pairs of words (such as IDOL-IDYL), or isolated triples such as OGEE-OGRE-OGLE or the distinct case IDLE-IDLY-ISLE, etc. The largest of the offshore islands is the eight-word "atoll" {QUAD, QUAI, QUAY, QUID, QUIP, QUIT, QUIZ, QUOD}. (See figure.)

As to question 2: In my dictionary at least, the Change-a-Word puzzle that requires the greatest number of steps to solve is this one:


These two words lie at opposite ends of the dictionary, so to speak. I'll leave it to the reader to find a chain of words connecting them! A hint is available here, and the solution is available here.

P.S. One assumes that 15-letter words are not concentrated in a massive continent...but I haven't looked at those yet!