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On "logic" and "the next number" in a sequence (fwd)



Sorry to revive the Marilyn Vos Savant thread from last summer,
but some of you might find my recent "fan letter" of moderate
interest.

John

---------- Forwarded message ----------
Date: Mon, 16 Mar 1998 12:01:15 -0800 (PST)
From: John Mallinckrodt <ajmallinckro@csupomona.edu>
To: marilyn@parade.com
Cc: AJMALLINCKRO@csupomona.edu
Subject: On "logic" and "the next number" in a sequence

16 March 1998

Dear Marilyn,

I write regarding your answer [Parade Magazine, 15 March
1998] to Christopher Adams who asked for the number that
"logically follows the numbers 1823 and 3469 and 9287." I
was surprised that you failed to point out to your readers
that there *is* no logical answer to such a question; there
are only more or less aesthetically appealing answers.

You, for instance, interpreted the numbers in terms of
identically positioned sets of four sequential key presses
on a numeric keypad that is rotated by ninety degrees
between each number. Cute.

But I would point out that these numbers can just as well be
viewed as the first three numbers in the series obtained
from the function

f(x) = 4349 - 4612 x + 2086 x^2

where x takes on the values 1, 2, 3, ... With this
interpretation, the number that follows them is

f(4) = 19277.

Of course they are also the first three numbers in the
series obtained from the function

g(x) = 23626 - (239719/6) x + 21363 x^2 - (19277/6) x^3

where x takes on the values 1, 2, 3, ... With this
interpretation, the next number is

g(4) = 0.

Clearly, by adjusting the values of the coefficients in a
cubic polynomial, I can make *any* number the next number.

So, while your answer may be appealing to some, I suspect
that aliens from the planet dodec (who have six fingers on
each hand, do arithmetic in base twelve, and have laid out
their numeric keypads quite differently than Earthlings)
would strongly prefer a method like one of mine. Nevertheless,
the plain fact is that *no* finite sequence of numbers can
ever, by themselves, "logically" determine the next number
in the sequence.

I would suggest that you owe your readers a retraction (and
a valuable lesson in the process.)

Sincerely,

A fan
-----------------------------------------------------------------
A. John Mallinckrodt http://www.intranet.csupomona.edu/~ajm
Professor of Physics mailto:ajmallinckro@csupomona.edu
Physics Department voice:909-869-4054
Cal Poly Pomona fax:909-869-5090
Pomona, CA 91768-4031 office:Building 8, Room 223