- #1
phlegmy
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[SOLVED] The four color map??
so i guess most of you know about the four color map theorm.
i read about it in a book a couple of days ago and had some idle brain time today while driving a tractor.
i scribbled out some maps on the dirt on the windows and always seems to enclose one region of the map in my attempt to find a five color map.
i then reasoned that if a five color map existed then there must be a section of it containing 5 mutually adjoining regions. and that if i could stand in anyone of the regions i should be able to walk to any of the other four without crossing another. and so i thought.. i'll draw dots to represent regions and lines to represent the path i'd take from one to another. so in joining the dots i again come to a stop, it is impossible for me to join all dots to all dots without crossing lines, i.e. distroying boarders between regions. so i thought, i'll draw another five dots and this time be carefull not to cross lines, but i end up in a situation where in order to connect two dots i must enclose a "deficient" dot.
so i figure there can never exist 5 mutally adjoining regions unless i can connect all the dots without crossing lines. if there and only 4 mutally adjoining regions [which can be easily done with dots and lines] i only need 4 colours.
i then wonder, does the orientations of the dots matter?, and i think, no, because i can alter my path,
i then wonder does it matter what order i connect the dots in, and i think, perhaps, but if the orientation of the dots isn't important then they may as well be points on a circle and so be very symetrical, so the order won't be too important
then i feel very happy that i will no longer waste my time trying to draw a five color map on the side of a trailer or the inside of a tractor cab. I've satisfied myself it cannot be done. then i wonder,.. does that consititute a proof.
so i do some googleing and find that some equally clever people have already done pretty much the same thing. but they don't call it proof??
so I'm figureing, I'm satisfied i cannot connect all the 5 dots, but i havnt "prooved" it. it's pretty obvious to someone who tries they cannot do it. [much more obvious than trying to draw funny maps!) so why is it not accepted as a proof?? or has anyone any thougths on this
apologies as always for my spelling
so i guess most of you know about the four color map theorm.
i read about it in a book a couple of days ago and had some idle brain time today while driving a tractor.
i scribbled out some maps on the dirt on the windows and always seems to enclose one region of the map in my attempt to find a five color map.
i then reasoned that if a five color map existed then there must be a section of it containing 5 mutually adjoining regions. and that if i could stand in anyone of the regions i should be able to walk to any of the other four without crossing another. and so i thought.. i'll draw dots to represent regions and lines to represent the path i'd take from one to another. so in joining the dots i again come to a stop, it is impossible for me to join all dots to all dots without crossing lines, i.e. distroying boarders between regions. so i thought, i'll draw another five dots and this time be carefull not to cross lines, but i end up in a situation where in order to connect two dots i must enclose a "deficient" dot.
so i figure there can never exist 5 mutally adjoining regions unless i can connect all the dots without crossing lines. if there and only 4 mutally adjoining regions [which can be easily done with dots and lines] i only need 4 colours.
i then wonder, does the orientations of the dots matter?, and i think, no, because i can alter my path,
i then wonder does it matter what order i connect the dots in, and i think, perhaps, but if the orientation of the dots isn't important then they may as well be points on a circle and so be very symetrical, so the order won't be too important
then i feel very happy that i will no longer waste my time trying to draw a five color map on the side of a trailer or the inside of a tractor cab. I've satisfied myself it cannot be done. then i wonder,.. does that consititute a proof.
so i do some googleing and find that some equally clever people have already done pretty much the same thing. but they don't call it proof??
so I'm figureing, I'm satisfied i cannot connect all the 5 dots, but i havnt "prooved" it. it's pretty obvious to someone who tries they cannot do it. [much more obvious than trying to draw funny maps!) so why is it not accepted as a proof?? or has anyone any thougths on this
apologies as always for my spelling