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Using Minesweeper to Solve NP
Blue Leader writes "Boston.com is reporting that the answer to one of math's most vexing problems lies in Minesweeper. Figure it out, and render modern encryption useless." Its a discussion of NP/P, as well as an excuse to play minesweeper. Personally, I kinda prefer mahjongg or tetris tho ;)
It's worth pointing out that the Boston.com does gloss over some fairly important mathematics.
All that Kaye has done is show that Minesweeper is NP complete. He has not yet found a polynomial-time solution to it, which is necessary to prove that P=NP -- in a nutshell, he just shows that Minesweeper falls into an equivalence class that holds a hell of a lot of other algorithms.
And EVEN IF HE FINDS the polynomial solution to Minesweeper, that STILL doesn't say anything about RSA (or any other "hard" algorithm), other than that it can be solved in polynomial time SOMEHOW.
The only reason people focus on this conjecture is they hope that proving P=NP and solving some algorithm will give them some magic insight into speeding up some other algorithm that's equivalently hard, rather than working on the algorithm directly. Or, disproving P=NP once and for all, and ensuring the computational assumptions that make people pick algorithms like RSA.
...was included with windows to give you something to favourably compare the 'bomb' rate to.
Comparisons:
Minesweeper:
- often explodes on the first click
- randomly explodes later on
- game is over quite quickly
- you have to press the reset button to restart
Windows:
- often explodes on the first click
- randomly explodes later on
- game is over quite quickly
- you have to press the reset button to restart
Its the same program!
Therefore- the Stability of Windows is NP complete! QED!
-WolfWithoutAClause
"Gravity is only a theory, not a fact!""History doesn't repeat itself, but it does rhyme." Mark Twain
The author of this paper has a web page here:
:)
He also has a page specifically about this Minesweeper business here.
I like the other paper proving that minesweeper, with a little variation, on an infinite board, is Turing-complete. Fun!
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- Give a man a fire and he's warm for a day, but set him on fire and he's warm for the rest of his life.
As with almost any mass media article about mathematics, this article is full of errors that nitpicky people like me feel the need to point out. First of all, some basic info you may be lacking. The basic P vs. NP problem is most simply stated as "P = NP?" P stands for Polynomial time, and NP stands for Nondeterministic Polynomial time, as in you can solve the problem is p(n) steps, where p is a polynomial and n is the size of the input file. Beyond that, some heinous mistakes they made: 1. P is a subset of NP, not a distinct set. Thus all P problems are NP (obviously, if you read the definition). 2. Internet encryption (at least RSA) is NOT KNOWN TO BE EVEN NP-COMPLETE. This is something I think a lot of people don't realize, and I have talked to many mathematicians who think that factorization will eventually be shown to be in P and thus RSA and all other such encryption schemes will collapse. All it takes is one brilliant hacker... 3. The answer has to do with determining consistency, which is very, very different from solving the game in a game theoretical sense. And some slightly more nitpicky issues: 1. NP-Complete problems are those problems whose solutions can be polynomial time transformed to solutions to _any_ other problem. That is why if you find a solution to the minesweeper problem, NP-Complete will cease to exist and P=NP. 2. No serious mathematician believes that P=NP. Anyone who wants to know more should read Sipser's book "Introduction to the Theory of Computation" which I highly recommend.
More details of the maths involved can be found at The ClayMath Institute's webpage and some related papers at R.W.Kaye's webpage
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I've always thought that one of the problems with math -- as taught at the secondary level -- was that you didn't actually learn any abstract math skills (well, you might, but you're not taught them). Just more algebraic manipulation, or, if you're luck, the foundations of calc.
I think games and optimization problems, though, could provide a fertile and interesting framework for teaching real mathematical thinking. Minesweeper. Knight's Tours. John Conway's games. Nim. Dominoes. Any small, discrete system with definable rules can get you thinking mathematically, though most people probably just play with heuristics....
Libertarianism is rich wolves and poor sheep playing gambler's ruin for dinner.
Besides P = NP for N = 1 (:-).
Unfortunately, this is not the minesweeper solving problem. It is the minesweeper consistency solving problem. 2 correct examples of the consistency problem:
X = MINE
O = COVERED EMPTY
NUMBER = BOARD CLUE
1 X
X X
The correct analysis of this board would be 'inconsistent'.
2 X
X O
The correct analysis of this board would be consistent.
The minesweeper consistency problem is a matter of checking the board and being able to declare whether or not the board is correct in all of its details.
The challenge is to construct a program which will process all generalized minesweeper boards and declare them correct/incorrect (accurately) in P. IF you can write such a program, then NP=P.
"Who is the Journal of Quantum Physics going to believe?" --Stephen Hawking
RSA is only NP, or at least nobody's proven it to be NP-Complete yet.
Any NP-Complete problem can be transformed into any other NP-Complete problem via a polynomial time transformation. Thus, solving one solves all. I have no idea how to do it, it's over my head. But it can be done.
Anyway, more to the point, this isn't about Minesweeper, it's a problem called the "MineSweeper Consistentcy Problem" and it's important to remember that. Essentially, the MCP is: given a half finished minesweeper board, is it consistent? That is, is it a valid board within the rules of the game? It is possible to get this board through normal play?
That's a bit of a different beast than just playing the game, guys.
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- Give a man a fire and he's warm for a day, but set him on fire and he's warm for the rest of his life.
When I was an undergraduate, I wrote a program that would detect whether a particular move in Minesweeper was safe. It used a recursive search, and couldn't detect safe moves in certain late-game situations where the mine count was relevant, but its play was otherwise perfect.
Since the program could definitively tell whether or not a move was safe, it could detect when a player was *GUESSING*. And so we could hack the program to always reveal a mine in such cases, driving the game weenies insane. :-)
Okay, we never built the search into the game, but we did hack Tetris in a few irritating ways... (As I understand it, Tetris is a lost game anyway: with probability 1, if you play long enough, you will lose, no matter your speed or strategy.)
The simple truth of the problem is that there is no one answer to it...
Not many people have a firm grasp of what this problem is really all about. Sure, you'll study it in your B.Sc or B.Tech...but really, even graduates fail to grasp some key concepts, although they study the tougher concepts....basically, this is how it goes:P is the set of problems that can be solved in deterministic polynomial time. That means for a problem with inputs of size N, there must be some way to solve the problem in F(N) steps for some polynomial F. F can be any polynomial, even N to the 10 millionth power.
NP is the set of problems you can solve in non-deterministic polynomial time. That means for a problem with inputs of size N, there must be some way to solve the problem in F(N) steps for some polynomial F just as before. In NP, however, the student is allowed to make lucky guesses, though it must prove the solution is correct. The standard format for a program in NP is: Guess the answer. Verify that the answer is correct in polynomial time. For example, factoring is in NP. Suppose you have a number A that you want to break into two factors. The NP program is: Guess factors P and Q. Multiply P times Q and verify that the results is A. This takes only two non-deterministic steps so the problem is in NP. Therefore, considering the differences between the two and the estimation involved, how is it possible to prove something like this?You can't "prove this". You can't disprove it either, but that's not the point - minesweeper is not going to help you with this.
Everything is but a number spoken by itself.
Wrong. It would do nothing of the kind. Proving Riemann's Zeta hypothesis would do that.
Even if you proved prime factorization of large numbers can be done in polynomial time, you would need an algorithm that accomplished it in a reasonable amount of time (seconds). An algorithm that had time complexity O(n^100) would still be polynomial, but useless in practice.