From: utzoo!decvax!harpo!ihps3!ihuxv!aark
Newsgroups: net.math
Title: Rebuttal: Using a computer to solve ABCDE problem (***SPOILER***)
Article-I.D.: ihuxv.190
Posted: Fri Jul 16 15:52:13 1982
Received: Sat Jul 17 02:39:33 1982


Why is using a computer to solve the ABCDE problem better
than using pencil, paper, and human reasoning?  Because it
has serendipitous results far beyond simply finding the
answer to the problem.

The question was, "What five-digit number ABCDE, when multiplied
by 4, gives the number EDCBA, with the digits in reverse
order?"  The answer is 21978.  I admit, this problem is easy to
do without a computer, simply by reasoning.  The computer technique
of trying all five-digit numbers until the answer is found
seems at first glance crude, cop-outish, and brute-force-ish in
comparison.

But!

Let's ask a few more questions, starting from the stated problem.

1.  "How many five-digit numbers ABCDE, when multiplied by any
other digit F, yield the answer EDCBA?"  Obviously, when F is 1,
any five-digit palindromic number works.  Let us eliminate these
trivial solutions and ask, "How many five-digit numbers ABCDE,
when multiplied by any digit F from 2 to 9, yield the answer EDCBA?"
If one had to rely on one's human abilities alone, many of us
would say, "That's too difficult" and drop the line of inquiry.
But a simple addition to the computer program allowed me to
instantly discover that the pair (21978, 4) is the ONLY pair that
satisfies the stated conditions.  Intuition gives no hint that
this is so.

2.  Unique things intrigue me, which prompted me to ask the next
question:  "For any positive integer n>=2, how many n-digit numbers
ABC..., when multiplied by any digit F from 2 to 9, yield the
answer ...CBA (same digits in reverse order)?"  Once again, a
simple addition to the exhaustive-search computer program
allowed me to experiment.  It turns out as follows:

	For n=2, no pair satisfies the conditions.
	For n=3, no pair satisfies the conditions.
	For n=4, the unique solution is (2178, 4).
	For n=5, the unique solution is (21978, 4).
	For n=6, the unique solution is (219978, 4).

Aha!  A pattern is beginning to form.  Could it be true that the
number 219...978, with any number of 9's, when multiplied by 4,
yields the reverse-order digits 879...912?  A quick check using
the calculator program on the computer proved the conjecture was
true for 2199978, 21999978, and 219999978.  This motivated me
to finally take pencil and paper and prove that the conjecture
is true for any number of 9's.  (Prove it yourself; it's easy
enough.)

3.  The last conjecture, which is suspect is true but haven't
tackled yet, is: "For any positive integer n>=2, the pair
(219...978, 4) is the ONLY pair of numbers (the first having n digits
and the second being >= 2 and <= 9) which, when multiplied together,
yields the first number with its digits reversed."  Any of you
pencil-pushers care to take that one on?

The point of all this is that without that powerful problem-
solving tool, the computer, I (and I would venture to say,
most of the rest of you) would never have gone on to discover
the more general truth that 219...978 times 4 equals 879...912
for any number of 9's.  Using the computer has sparked my
imagination and given serendipitous results.

Of course, this result is not very useful; my life has not
been radically changed by it; it will not solve the problems
of the world.  However, it's the principle that's important.
Those narrow-minded superior types who eschew "brute-force"
computer solutions and denigrate those who use the computer
in this way will, I assert, never discover the things that we
enlightened computer users do.  Remember, no one was able to
prove the Four Color Map theorem until they brought in a computer
to aid human intuition and do the dirty work in a manner similar
to that described above.

Alan Kaminsky
... ihps3!ihuxv!aark

P.S.  Those who feel compelled to continue this discussion,
let's move it to the net.followup newsgroup, please.  I'll be
happy to go on arguing, but let's leave net.math uncluttered.