You're on the right track, Pat, especially with your second point. Computers can't prove anything. Basically, because they can't do real math (excuse the pun). They only deal with finite numbers, very big finite numbers maybe, but still finite. And there are two types of infinite sets, countable (like rationals that can be put into 1-to-1 correspondence to integers) and uncountable (like irrationals that can't, or reals that are a set with both rational and rational numbers). Robert -----Original Message----- From: [log in to unmask] [SMTP:[log in to unmask]] Sent: Monday, March 12, 2001 1:33 PM To: [log in to unmask] Subject: Re: (VERY) OFF TOPIC - Map coloring In a message dated 3/12/01 12:14:55 PM Eastern Standard Time, [log in to unmask] writes: It is the job of the mathematicians is to determine what is mathematically meant by "every case". That's done all the time. For example, in poker, the "odds" of getting five cards of the same suit is determined by calculating the NUMBER of ways five randomly drawn cards can be of the same suit, and then dividing that number by the TOTAL NUMBER OF WAYS any five cards can be drawn from a 52 card deck. Yes, and if n colors are going to be used to color n shapes, the number of possibilities is n to the nth power. This really includes all possibilities, including using only one of the n colors for all of the n shapes. Most of these color arrangments are not going to meet the criteria for what Saaty calls correct k-coloring--that is, no two contiguous countries are the same color. For his correct k-coloring, the limit is n!. That is, n factorial. Thus, for four color used on a basic four-shape module, there are 256 possible combinations of which only 24 are "correct." Now here are my questions. 1) This sounds to me like it's leading into a problem in what I believe is called combinatorial algebra. So why does one need a computer? 2) For each n in the number series, the nth power of n is an infinite series. So is n factorial. So the computer, if using brute force, is going to run out of steam at some point, as it can't deal with an infinite series by testing "every" possibility. 3) What was wrong with de Morgan's proof, which was rejected? It seems to me that he was on the right track, much more so than what followed after. 4) Maybe you should rethink your poker example. The number is finite because there's a limit to the number of cards in a pack. There's no limit to the number of shapes that can be on a map. That's why one has to do something along the line of breaking a large map down into, say, units of four or fewer shapes. How, exactly, did they compute the number of possible combinations, and what did the number turn out to be? You keep assuring me it was "gazillions," which I don't find very clear. I'm actually more interested in how it was computed than what it turned out to be. pat