Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.2 9/18/84; site x.UUCP
Path: utzoo!linus!philabs!cmcl2!seismo!harvard!think!mit-eddie!cybvax0!frog!x!john
From: john@x.UUCP (John Woods)
Newsgroups: net.puzzle
Subject: Re: x to the x to the x ... (maybe SPOILER)
Message-ID: <477@x.UUCP>
Date: Thu, 25-Apr-85 10:29:15 EDT
Article-I.D.: x.477
Posted: Thu Apr 25 10:29:15 1985
Date-Received: Sun, 28-Apr-85 04:24:35 EDT
References: <748@whuxlm.UUCP>
Distribution: net
Organization: Charles River Data Systems, Framingham MA
Lines: 35

> This one may be old, it's from an old MAA test.
> x^x^x^x... = 2
> the x's go on forever and the carots mean "power of", not "exclusive OR".
> What is the value of x?
> 
I remembered the form of the answer but not the actual answer, so I guess
I half-cheated.  It's OK, I'm only half confident in my answer.

	x^x^x^x^... = 2
          A
	  |
	  ---
	    |
Let zeta be x^x^x^x^..., and re-write the left-hand side as

	x^zeta = 2	Note also that x^zeta = zeta, and therefore that
			2 = zeta.  Rewrite as
	x^2 = 2

Simple, no?  Not done yet:  there are two answers, plus and minus root-2.

It (seems to me that it) can't be plus root-2, because trying to do the
"series" by hand leads to an unbounded value.  So it must be minus root-2.
However, I am not comfortable with that, either, because I don't remember
my complex arithmetic well enough to work out whether or not it works (I
knew it - I've been programming in C too long!  I had to look in a FORTRAN
manual just to convince myself that log and exponent operations are OK on
negative quantities.  All that math in college has just poured out my ears!).

Well, I tried.
-- 
John Woods, Charles River Data Systems, Framingham MA, (617) 626-1101
...!decvax!frog!john, ...!mit-eddie!jfw, jfw%mit-ccc@MIT-XX.ARPA

You can't spell "vile" without "vi".