From: Chip Eastham Subject: Re: Non-computer proof of 4 Color Theorem Date: Sun, 15 Oct 2000 15:05:01 GMT Newsgroups: sci.math,alt.math.recreational Summary: [missing] In article <8satbp$i5$1@salmon.maths.tcd.ie>, tim@maths.tcd.ie (Timothy Murphy) wrote: > Chip Eastham writes: > > >There is a nicely presented approach to proving the Four Color Theorem > >without any computer help at the following Yahoo! Geocities Web site: > > >http://www.geocities.com/dharwadker/index.html > > Without criticising the above in any way, > it is worth noting that the original proof of the 4-colour theorem > did not actually use the computer in the _proof_, IIRC. > It was only used to _predict_ the best route to take in constructing the proof. > All the (partial) maps that arose were in fact carefully drawn by hand > (in the preprint version) and presumably checked the same way. > > -- > Timothy Murphy > e-mail: tim@maths.tcd.ie > tel: 086-233 6090 > s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland > Timothy, The proof originally presented by Appel and Haken has two parts, showing that a certain collection of 1476 configurations (maps) are all "reducible" and showing that the same collection is "unavoidable". By reducible is meant a certain technical characteristic that rules out their being contained in a minimal counterexample to four-coloring of planar maps, while unavoidable means that any minimal counterexample must contain one of the these configurations. The part of their proof that could be checked by hand is the reducibility of each of the 1476 configurations; the proof of unavoidability depends on a discharging algorithm they formulated with complex logic and which could only be carried out by computer. In 1996 Robertson, Sanders, Seymour, and Thomas presented an approach to the Four Color Theorem which uses a smaller collection of 633 configurations and makes other simplifications. A summary and links to further downloads are given at: http://www.math.gatech.edu/~thomas/FC/fourcolor.html Although their proof still has an essential dependence on computer programming in demonstrating the unavoidable set of 633 maps, they state that the hand-checking of reducibility can be carried out in about two months (or in 20 minutes by computer!). Regards, Chip Sent via Deja.com http://www.deja.com/ Before you buy.