From: Ruby Quiz <james@...>
Date: 2007-09-06T21:00:20+09:00
Subject: [QUIZ] Count and Say (#138)

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by Martin DeMello

Conway's "Look and Say" sequence
(http://en.wikipedia.org/wiki/Look-and-say_sequence) is a sequence of numbers in
which each term "reads aloud" the digits of the previous term. For instance, the
canonical L&S sequence starts off 1, 11, 21, 1211, 111221, ..., because:

	* 1 is read off as "one 1" or 11.
	* 11 is read off as "two 1's" or 21.
	* 21 is read off as "one 2, then one 1" or 1211.
	* 1211 is read off as "one 1, then one 2, then two 1's" or 111221.
	* 111221 is read off as "three 1, then two 2, then one 1" or 312211.

Over on rec.puzzles, Eric A. proposed a variant in which the letters of a
sentence are grouped, then "counted aloud", omitting the "s"s for the plural
form. Thus, seeding the sequence with "LOOK AND SAY", we get:

	0. LOOK AND SAY
	1. TWO A ONE D ONE K ONE L ONE N TWO O ONE S ONE Y
	2. ONE A ONE D SIX E ONE K ONE L SEVEN N NINE O ONE S TWO T TWO W ONE Y
	3. ONE A ONE D TEN E TWO I ONE K ONE L TEN N NINE O THREE S THREE T ONE V
	   THREE W ONE X ONE Y

and so on. (Note the difference between this and the L&S sequence--the letters
are counted rather than read in order). Eric wants to know when the sequence
enters a cycle, and how long that cycle is. Well?