But basically problems where you have to take the whole grid into place are difficult to solve, and there are peculiar positions that can really only occur in large grids that would work like logic gates which makes it even harder to solve quickly
I think I’m too dumb for anything beyond like halfway down page 3 in that paper.
I guess maybe the problem is that I don’t know what “solved” means in this context (especially in regards to it’s 99.99% solved, but not 100% solved).
Like I know, say, tic-tac-toe is solved, because there’s a certain set of moves that can guarantee a draw every time. Is it like that? I assume so, but there’s some connection I’m not getting.
Yeah I admit I don’t fully understand the problem either when you start getting into the insane details.
Well like the other example I’ve heard of is the traveling salesman problem. Say you have a salesman that wants to travel between 200 cities. There is an efficient algorithm that is within 99% of the optimal results. But if you want the optimal result you have to brute force it which is not efficient at all.
If you have a random minesweeper position, it’s very likely that you can efficiently figure out where the mines are (or draw as many inferences about them as possible). But there are a small number of pathological positions where there is enough information to figure out where the mines are, but there’s no algorithm that can work it out efficiently.
Although, I don’t really understand how they are going to build low income housing there if it costs $8m an acre, unless it’s just some rich person doing charity.
Man, just put a tax on single-family zoned areas. People can either pay the tax to keep the riff-raff out or let people in. Take the money raised and use it to pay for building low income housing.
The best approximations aren’t quite that good. The best known approximation for TSP can be as bad as 150% of optimal: Christofides algorithm - Wikipedia
Also, more importantly, you don’t need to brute force TSP to get an optimal solution. Brute force is O(n!), which means you could basically never solve anything. P != NP just requires that it take exponential time, and indeed the Held-Karl algorithm (and others) can give an optimal solution in O(2^n), which is vastly faster than brute force (e.g., 20! ~= 1e18, totally unsolvable, while 2^20 is roughly a million, and trivial to do).
When someone was wrong on the internet in the 19th century:
The Victorian-age mathematician, logician, and writer Charles Lutwidge Dodgson, better known by his pseudonym Lewis Carroll, also expressed interest in debunking illogical circle-squaring theories. In one of his diary entries for 1855, Dodgson listed books he hoped to write, including one called “Plain Facts for Circle-Squarers”. In the introduction to “A New Theory of Parallels”, Dodgson recounted an attempt to demonstrate logical errors to a couple of circle-squarers, stating:[40]
The first of these two misguided visionaries filled me with a great ambition to do a feat I have never heard of as accomplished by man, namely to convince a circle squarer of his error! The value my friend selected for Pi was 3.2: the enormous error tempted me with the idea that it could be easily demonstrated to BE an error. More than a score of letters were interchanged before I became sadly convinced that I had no chance.