
Mathematicians Solve Minimum Sudoku Problem - llambda
http://www.technologyreview.com/blog/arxiv/27469/
======
yread
7.100.000 hours on a 640 hexcore Xeon cluster is only 1.848 hours. That is ~6
hours a day for January-December.

I always had a suspicion that all these super powerful grids are only used to
play Sudoku..

------
supar
"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.

~~~
pgmcgee
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#Games_and_puzzles)

~~~
Someone
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.

~~~
pbhjpbhj
Surely a 1x1 grid requires 0 clues?!

~~~
Someone
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?

~~~
T-hawk
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.

~~~
Someone
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.

------
Almaviva
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?

~~~
its_so_on
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?

~~~
SatvikBeri
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.

~~~
its_so_on
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.

------
joshuahedlund
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.

------
arctangent
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_computer)

~~~
dxbydt
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.

~~~
nandemo
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>

------
jakewalker
It's funny because the way I play Sudoku is often through sheer brute force as
well.

~~~
fennecfoxen
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> )

~~~
Tossrock
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.

------
nradov
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>

------
ctdonath
Sudoku: crack for intellectuals.

------
stevewillows
I'm glad we've got our top minds on this.

