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You never need more than four colors to color every country on a map a different color from its neighbours. This was proved in the 20th century — but nobody knows why it is true. Heh. I suppose we've proved it, but this author doesn't accept this proof as an explanation for why it's true.
The "proof" you're most likely talking about doesn't show why this is true. Merely shows that it is true. Exhaustion doesn't qualify as a real argument, in my opinion.
This is a game? It takes maybe 2 seconds to find a solution on this thing.
Y'all did not invent a d20!
This is just a really nifty site. Much deeper than I know how to express. One image per page, but it's really good. ("Bookmarked", so to speak.)
Made me dig up this old guy: https://www.google.com/search?q=5+%2B+(-sqrt(1-x%5E2-(y-abs(...
http://images-of-math.tumblr.com/post/105433933328/the-goal-... This is a game? It takes maybe 2 seconds to find a solution on this thing.
You never need more than four colors to color every country on a map a different color from its neighbours. This was proved in the 20th century — but nobody knows why it is true. Heh. I suppose we've proved it, but this author doesn't accept this proof as an explanation for why it's true.
The "proof" you're most likely talking about doesn't show why this is true. Merely shows that it is true. Exhaustion doesn't qualify as a real argument, in my opinion.
You never need more than four colors to color every country on a map a different color from its neighbours. This was proved in the 20th century — but nobody knows why it is true. Heh. I suppose we've proved it, but this author doesn't accept this proof as an explanation for why it's true.
Is that true in today's geography? I know of one almost [1] counterexample (of that specific statement, not of the four-color theorem): the North Sea, Belgium, the French Republic, Germany and the Kingdom of the Netherlands all border each other, so you need five different colors to color them on a map.
[1] Almost because the Kingdom of the Netherlands isn't a country.
Confused? Read http://en.m.wikipedia.org/wiki/Kingdom_of_the_Netherlands, in particular the part about overseas territories, and notice that Sint Maarten borders Saint-Martin.
[on an even more sideways track: the US dollar is an official currency in part of the _country_ of the Netherlands (in Bonaire, Saba and St Eustatius)]
Back to my original question: does anybody know of a valid counterexample for the statement on countries?
You never need more than four colors to color every country on a map a different color from its neighbours. This was proved in the 20th century — but nobody knows why it is true. Heh. I suppose we've proved it, but this author doesn't accept this proof as an explanation for why it's true.
"You never need more than four colors to color every country on a map a different color from its neighbors" Is that true in today's geography? I know of one almost [1] counterexample (of that specific statement, not of the four-color theorem): the North Sea, Belgium, the French Republic, Germany and the Kingdom of the Netherlands all border each other, so you need five different colors to color them on a map. [1] Alm…
Neither is the North Sea! Nonetheless, this is a neat example.
> Back to my original question: does anybody know of a valid counterexample for the statement on countries?
A standard counterexample to the hypotheses of the 4-colour theorem (though not to the conclusion, as consulting a map easily verifies) is Michigan, which is not connected.
The 4-colour theorem's hypotheses also rules out the possibility of 4 countries meeting at a corner (or, rather, declare that they don't meet in that case). If there were such an arrangement—and I'd be surprised if there isn't; for 3 countries, one has the example of Finland, Sweden, and Norway—then it would be easy to juice it up to a counterexample.
Maybe the guy who established an island with a bizarre currency, including one coin that had a denomination of π (I can't remember who—I thought Dean Kamen, but his Wikipedia page doesn't mention it), could be induced to subdivide his island in such a way as to create a counterexample. :-)
Earlier quoted context omitted.
The "proof" you're most likely talking about doesn't show why this is true. Merely shows that it is true. Exhaustion doesn't qualify as a real argument, in my opinion.
Personally, I think it's totally valid. (PhD mathematician here).
Earlier quoted context omitted.
The "proof" you're most likely talking about doesn't show why this is true. Merely shows that it is true. Exhaustion doesn't qualify as a real argument, in my opinion.
Personally, I think it's totally valid. (PhD mathematician here).
To be fair, though it's a bit ambiguous, I read sebastialonso's post https://news.ycombinator.com/item?id=8799362 as indicating that he or she believes that the result is true—i.e., does not necessarily reject the validity of proof by exhaustion as a style of argumentation—but does not feel that the proof by exhaustion is an explanation.