Mathematicians Solve Minimum Sudoku Problem(technologyreview.com)
technologyreview.com
Mathematicians Solve Minimum Sudoku Problem
http://www.technologyreview.com/blog/arxiv/27469/
11 comments
"These guys have solved the problem using the tried and trusted mathematical technique of sheer brute force.
In essence these guys have examined every potential 16-clue solution for every possible Sudoku grid."
I was actually hoping for a formal solution or proof here by reading the title.
In essence these guys have examined every potential 16-clue solution for every possible Sudoku grid."
I was actually hoping for a formal solution or proof here by reading the title.
They did produce a formal solution/proof. The disappointing part is that it contains billions of statements that don't give us any insight about the problem.
I was also hoping to see some mathematical insight that allowed them to prove the minimal sudoku problem. The problem with a brute force solution is that it only applies to the 9x9 sudoku. It would be interesting to see what the limit is for larger sudokus and what correlations there are between their dimensions and their minimal number of clues.
I believe that would have far-reaching implications, seeing that Sudoku has been proven to be an NP-Complete problem.
http://en.wikipedia.org/wiki/List_of_NP-complete_problems#Ga...
http://en.wikipedia.org/wiki/List_of_NP-complete_problems#Ga...
Proving something about a NxN grid does not have much to do with a problem being NP-complete.
For example, the fact that it is not very hard to proof that a 1x1 sudoku requires 1 clue does not prove that sudoku is not NP-complete.
For example, the fact that it is not very hard to proof that a 1x1 sudoku requires 1 clue does not prove that sudoku is not NP-complete.
Surely a 1x1 grid requires 0 clues?!
A classic off-by-one error. That is not what I was thinking of, though. A 2x2 grid requires a single clue.
A 3x3 grid requires 2 (1 and 2 across a diagonal suffices)
So, we have a sequence 0,1,2,?,?,?,?,?,17,...
I checked the OEIS, but it does not seem to contain this sequence. Does anybody know more values?
A 3x3 grid requires 2 (1 and 2 across a diagonal suffices)
So, we have a sequence 0,1,2,?,?,?,?,?,17,...
I checked the OEIS, but it does not seem to contain this sequence. Does anybody know more values?
It's not really valid to treat this as a sequence, since most side lengths don't form what we think of as a Sudoku puzzle. The side length of a true Sudoku must be a perfect square in order to create the interior boxes that are (geometrical) squares. For the counterexample, think of a 7x7 puzzle (or any prime)... how can a square box contain seven cells?
A 4x4 puzzle gives you four 2x2 boxes. The standard 9x9 puzzle gives nine 3x3 boxes. A 16x16 puzzle gives sixteen 4x4 boxes. Sudoku variations sometimes have rectangular boxes; a 6x6 puzzle can have six 2x3 boxes, or a 12x12 puzzle can have twelve 3x4 boxes. But that is a different form of constraint logic so we shouldn't expect that the minimum number of clues for these sizes would follow a recognizable sequence.
A 4x4 puzzle gives you four 2x2 boxes. The standard 9x9 puzzle gives nine 3x3 boxes. A 16x16 puzzle gives sixteen 4x4 boxes. Sudoku variations sometimes have rectangular boxes; a 6x6 puzzle can have six 2x3 boxes, or a 12x12 puzzle can have twelve 3x4 boxes. But that is a different form of constraint logic so we shouldn't expect that the minimum number of clues for these sizes would follow a recognizable sequence.
I had to check Wikipedia (which, as we all know, always is right :-)) to see that you are right. I have seen so many variations on sudoku's that I forgot what the original looked like. I was just thinking of Latin squares.
No because then you wouldn't be able to arrive at a fixed solution there would be 9 solutions with 0 clues.
In a 1x1 grid you would only use one digit, not 9. A 1x1 grid requires 0 clues, a 2x2 requires 1 clue.
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(tongue in cheek) right, a 1x1 grid requires 0 clues, it has only a single digit as its solution; that digit is -- [its_so_on looks left, cut to SatvikBeri]
Suppose there is a unique 15-clue solution. Then add another clue by adding any number from the unique solution. The solution must still be unique (because it contains the 15 clues), so we now have a unique 16-clue solution, which is impossible.
So isn't this article missing the obvious?
So isn't this article missing the obvious?
I think they addressed this in the edit they made:
>Correction: this post was edited on the 6 January to reflect the argument that if an n-clue grid is uniquely solvable then adding a digit to make an n+1-clue grid must also be uniquely solvable. So if there are no uniquely solvable 16-clue grids, there cannot be any grids with fewer clues that are uniquely solvable. Thanks to RealMurph and abooij.
>Correction: this post was edited on the 6 January to reflect the argument that if an n-clue grid is uniquely solvable then adding a digit to make an n+1-clue grid must also be uniquely solvable. So if there are no uniquely solvable 16-clue grids, there cannot be any grids with fewer clues that are uniquely solvable. Thanks to RealMurph and abooij.
The implication (not entirely clearly stated in the article) is that if a puzzle with n clues is solvable, a puzzle with n+1 clues is also solvable.
Conversely, if a puzzle with n clues is unsolvable, any puzzle with n-1 clues is similarly unsolvable.
Conversely, if a puzzle with n clues is unsolvable, any puzzle with n-1 clues is similarly unsolvable.
I don't get where the article makes this assumption:
"It's easy to see why. A grid with 7 clues cannot have a unique answer because the two missing digits can always be interchanged in any solution."
let's say these are your 7 clues: (row 1) 123 456 7
the article suggests that IF there is a full solution with 123 456 789 then there is a full solution with 123 456 798
but what guarantees that the second (or first) of these 9 clues don't lead to an "impossible" scenario, so that only one of the two is actually possible to solve?
Maybe I am misunderstanding the article.
Turning to what you say: "Suppose there is a unique 15-clue solution. Then add another clue by adding any number from the unique solution. The solution must still be unique (because it contains the 15 clues), so we now have a unique 16-clue solution, which is impossible."
Can you elaborate on why a unique 16-clue solution is impossible?
"It's easy to see why. A grid with 7 clues cannot have a unique answer because the two missing digits can always be interchanged in any solution."
let's say these are your 7 clues: (row 1) 123 456 7
the article suggests that IF there is a full solution with 123 456 789 then there is a full solution with 123 456 798
but what guarantees that the second (or first) of these 9 clues don't lead to an "impossible" scenario, so that only one of the two is actually possible to solve?
Maybe I am misunderstanding the article.
Turning to what you say: "Suppose there is a unique 15-clue solution. Then add another clue by adding any number from the unique solution. The solution must still be unique (because it contains the 15 clues), so we now have a unique 16-clue solution, which is impossible."
Can you elaborate on why a unique 16-clue solution is impossible?
For the first question, about why 123 456 7 doesn't have a unique solution:
Let's say there is a unique solution to 123 456 789 . In that solution, swap every 8 and 9. It's not hard to see that this will be a correct solution of 123 456 798. Therefore, 123 456 7 must have an even number of solutions.
For the second question, about why a unique 16-clue solution is impossible: that's the result mentioned in the article, with the proof that took a year to calculate.
Let's say there is a unique solution to 123 456 789 . In that solution, swap every 8 and 9. It's not hard to see that this will be a correct solution of 123 456 798. Therefore, 123 456 7 must have an even number of solutions.
For the second question, about why a unique 16-clue solution is impossible: that's the result mentioned in the article, with the proof that took a year to calculate.
For the first question: thank you, I didn't realize that the article meant that those two numbers are supposed to be swapped THROUGHOUT! (every occurrence in the grid, like find-and-replace). This makes sense to me. Likewise, it seems to me true that you can swap-and-replace any two numbers in any completed graph. (Really, they're just symbols, it could be turning the original numbers into A B C D E F G H I in the first step - then you can map these 1:1 to one through nine in the second step however you want) and, therefore, you could do that operation on the original clues and get a valid sudoku as well.
In other words, it seems if you see some sixes and some fives in a sudoku, you can just swap them before you solve them, getting a different, but still valid sudoku. Interesting.
For the second question, thanks. I thought your parent meant something different - that the 16-clue solution was "obviously" impossible.
In other words, it seems if you see some sixes and some fives in a sudoku, you can just swap them before you solve them, getting a different, but still valid sudoku. Interesting.
For the second question, thanks. I thought your parent meant something different - that the 16-clue solution was "obviously" impossible.
Regarding 7 hints provided. Suppose you have a puzzle which contains 7 digits for hints. Assumming the provided digits are 1-7 placed around the grid and that they permit you to produce a unique solution for those digits (that is you can fill up all but 18 spaces). If you then fill in those remaining spaces with 9 8s and 9 9s, then, however you arranged them, the opposite arrangement is valid.
Essentially those two remaining hints would be variables that you could then substitute either symbol for yielding 2 potential solutions making it an invalid sudoku puzzle.
Essentially those two remaining hints would be variables that you could then substitute either symbol for yielding 2 potential solutions making it an invalid sudoku puzzle.
thanks. it's like why a system of three variables for which you have two equations (like x = 2y + z and x = 3y + 2z + 7) isn't solvable.
It's interesting that we are reaching an era where we can prove things through computational brute force before coming up with an elegant proof, even if it means running computers for 12 months.
I wonder what implications this has for other mathematical complexities. Obviously it helps that Sudoku involves concepts that are very discrete... you can't really do brute force regarding the number line... but there have to be other discrete issues. (For example, every now and then I have the occasional fleeting terror that a sudden technology breakthrough will instantly render all pre-existing SSL encryption worthless.)
Back to the Sudoku, I'm also now interested in how many 17-clue solutions there are, and how many 18-clue, and if there's any sort of pattern.
I wonder what implications this has for other mathematical complexities. Obviously it helps that Sudoku involves concepts that are very discrete... you can't really do brute force regarding the number line... but there have to be other discrete issues. (For example, every now and then I have the occasional fleeting terror that a sudden technology breakthrough will instantly render all pre-existing SSL encryption worthless.)
Back to the Sudoku, I'm also now interested in how many 17-clue solutions there are, and how many 18-clue, and if there's any sort of pattern.
This article reminded me of the computer-assisted proof of the Four-colour Theorem:
http://en.wikipedia.org/wiki/Four_color_theorem#Proof_by_com...
http://en.wikipedia.org/wiki/Four_color_theorem#Proof_by_com...
It reminded me of what Dr. Doron Zeilberger has been repeating ad-nauseam ( eg: http://www.math.rutgers.edu/~zeilberg/Opinion36.html )
Ultimately automated theorem provers are going to rule. Dr.Z actually takes the very radical stand of calling all paper-pencil math proofs of Paul Erdos trivial, whereas "Theorems that only computers can prove...are not quite as trivial". The fact that we have to reach out to a computer to assist in solving min-sudoko is unfortunate but at the same time, inevitable. Dr. Z mentions Four Color Theorem, Kepler's Conjecture, and Conway's Lost Cosmological Theorem as other theorems only computers can solve.
But note that the proofs for the 4-Color Theorem and this Sudoku theorem are "merely" computer-assisted. The part of reducing the Sudoku search space, for instance, seems to have been done pencil-and-paper style. What Zeilberger wants is completely automated proving.
Oh, and thanks for introducing me to Conway's Cosmological Theorem. Very interesting!
http://mathworld.wolfram.com/LookandSaySequence.html
Oh, and thanks for introducing me to Conway's Cosmological Theorem. Very interesting!
http://mathworld.wolfram.com/LookandSaySequence.html
It's funny because the way I play Sudoku is often through sheer brute force as well.
In college I wrote a little Sudoku solver for an algorithms class which was more or less brute-force with backtracking (and a little intelligence about which part was the weakest and should be brute-forced first). A fun exercise, but for better or worse, I don't play Sudoku anymore. I solved it; the slowest part of getting the computer to do it for me now is the data entry. I've lost all motivation to do it by hand.
(Online + GPL'd: http://random.fennecfoxen.org/webtoys/sudo/sudo.html )
(Online + GPL'd: http://random.fennecfoxen.org/webtoys/sudo/sudo.html )
I wrote an intelligent solver (doing set elimination, etc) in Java back in high school, but couldn't get it to solve the harder puzzles. Clearly I was lacking one of the essential implications that allow you solve them manually. I actually never even considered brute forcing it.
This has been posted before, but for those who haven't read it Peter Norvig has an excellent article on Sudoku solving algorithms.
http://norvig.com/sudoku.html
http://norvig.com/sudoku.html
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I'm glad we've got our top minds on this.
I always had a suspicion that all these super powerful grids are only used to play Sudoku..