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Complete the sequence problems (fwd)



Dear Phys-L colleagues,

As Phys-L seems a little, shall we say, listless this weekend, I thought
I'd share a copy of a message I just sent to Ms. Vos Savant concerning an
item in her column that will appear tomorrow. (Wow! How did I do that?)

John

---------- Forwarded message ----------
Date: Sat, 17 Jul 1999 14:18:25 -0800 (PST)
From: John Mallinckrodt <ajmallinckro@csupomona.edu>
To: marilyn@parade.com
Cc: John Mallinckrodt <AJMALLINCKRO@csupomona.edu>
Subject: Complete the sequence problems

Marilyn,

Once again you have fallen into the trap, as you often do, of
supposing that the first n members of a sequence can determine the
(n+1)st member or beyond. Recently, a reader sent you the
sequence beginning

Y, Y, H, L, Y, E, Y, T, R

and asked for the next three members of the sequence. You
interpreted these values as the final letters of the first nine
months of the year and supposed that the next three members of the
sequence would be the final letters of the next three months--
"R,R,R"

But, however compelling that interpretation might be, it is simply
not dictated by the values themselves and the next three members
of the sequence could just as well be "A,A,A." For instance, if I
interpret the given sequence in terms of the associated numerical
positions in the alphabet, it becomes

25, 25, 8, 12, 25, 5, 25, 20, 18.

These numbers can, in turn, be interpreted as the values of some
function f(x) evaluated at x =

1, 2, 3, 4, 5, 6, 7, 8, 9

Of course there are an infinite number of such functions one of
which is

f(x) = 23455 - (1912504277/27720) x
+ (200368663/2400) x^2 - (1446068941/25920) x^3
+ (2802040471/120960) x^4 - (4598563451/725760) x^5
+ (33809683/28800) x^6 - (35865479/241920) x^7
+ (72409/5760) x^8 - (70979/103680) x^9
+ (13081/604800) x^10 - (481/1596672) x^11

and it "turns out" that this particular function has the
"surprising" property that f(10) = f(11) = f(12) = 1 which would
all be interpreted as "A"s.

Obviously, by suitable choice of the numerical coefficients I can
make the next three members of the sequence be whatever I choose.
Furthermore, by altering the rules of interpretation I can
introduce numerical elements, punctuation marks, or whatever I
want into the sequence.

I'm sure you understand all of this, but I fear that you may enjoy
the game of "solving" reader supplied "complete the sequence
puzzles" just a little bit too much to explain to your readership
why they simply have no unique solutions.

Sincerely,

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