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.