Conference rusure::math

    The standard answer to this kind of argument runs along the lines
    that, sure, you can produce generating functions all over the place
    but the "right answer" is the one the smart people come up with.
    
    If you are given the sequence:  1,2,3,4,5,6,__ it is natural to
    assume that most people will answer 7.  Anybody answering "-1.1"
    is clearly just being precocious, and deserves to get docked.
    
    Another argument holds that Occam's Razor applies, i.e., among all
    the right answers, the simplest is best.
    
      John

    This type of number sequence problem was discussed at length in
    a "Mathematical Games" (or "Metamagical Themas"?) column of
    "Scientific American" earlier this year.  Can someone provide
    a tighter bound?

This CAN be defined mathematically (sometimes!) as finding the next
number, using such a "g" function with the minimal complexity...

ND

    re .1 -- Yes,  if there is a simple answer or you can find a definition
    of simple that applies to all possible problems.  In my opinion the
    notion of simple is relative and therefore such an argument is
    inapplicable to testing intelligence which is also relative.  For
    example the test compiler's idea of simple may be different to that
    of the lesser able candidate,  but more importantly,  it may be
    different to that of the candidate of very high intelligence.  As
    a result the intelligent persons idea of a simple answer will include
    a larger range of valid answers.
    Occam's razor was originally applied to Natural law hypotheses,  many
    of which were discovered to be too simple.  Indeed,  we have the
    more advanced razor of Einstein which,  translated into English,
    would be,  "The Creator is subtle but not mischievous."

    The other point is that the notion of obvious,  apart from being
    relative,  is context driven.  In fact I deliberately placed this
    problem after the previous one to enable the reader to benefit
    from a contextual path to an answer that,  although this path is
    subtle,  is definitely there.  What is obvious will generally
    depend upon what previous questions have just been answered.
    As de Bono pointed out,  the state of the brain is not analagous
    to a towel on which ink spots may be dropped in any sequence but
    in the same quantities at the same points to produce the same
    pattern,  but is more analagous to a jelly on which you drop hot
    ink,  so that the flow of ink will create channels dependent on
    the order in which you made the previous drops.  (Ref. "PO: Beyond
    Yes and No",  Edward de Bono)

    But my aim was in fact to present a mathematical problem for which
    I do have an answer,  rather than provoke philosophical discussion
    alone!

    Spoiler follows:
    
    		 n       i-1        n
    		---   	 ---	   ---
    		\	 | |	   | |
		/   T  ( | | (x-j) | | (x-j)
  		---  i
    		i=1	 j=1  	   j=i+1
    	y = 	------------------------
    		 n    i-1        n
    		---   ---	---
    		\     | |	| |
		/   ( | | (x-j) | | (x-j)
  		---
    		i=1   j=1  	j=i+1