## Sudoku Math Puzzle

A forum for good logic/math puzzles.

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factorialite
Posts: 16
Joined: Thu Nov 01, 2007 2:12 am UTC

### Sudoku Math Puzzle

Do we know the 9x9 Sudoku with the largest determinant? If so, what is said puzzle?

Do we know the 9x9 Sudoku with the largest 3x3 determinant, where the 3x3 numbers are the determinants of each 1-9 section?

I've just been curious.

quintopia
Posts: 2906
Joined: Fri Nov 17, 2006 2:53 am UTC
Location: atlanta, ga

### Re: Sudoku Math Puzzle

I'm sure you can quickly brute force the largest determinant value using only 1-9 for 3x3. From this, you'll know that any of the bajillions of puzzles which have a 3x3 section with this determinant value are solutions to your second question.

And since this doesn't seem to be a puzzle (by which I mean the solution is known and it is posted as a challenge to other users), perhaps it would go better on Math?

Cosmologicon
Posts: 1806
Joined: Sat Nov 25, 2006 9:47 am UTC
Location: Cambridge MA USA
Contact:

### Re: Sudoku Math Puzzle

It may be possible to brute-force the 9x9 as well, at least part way. There are 5,472,730,538 unique sudoku if you count the symmetries as identical. Some of these symmetries - like swapping a row or column - leave the magnitude of the determinant invariant, but some - like interchanging all of two digits - don't. If you could come up with a way to find the maximum determinant for all symmetric variations of a given sudoku in under, say, a second, that would bring it into the realm of possibility.

factorialite
Posts: 16
Joined: Thu Nov 01, 2007 2:12 am UTC

### Re: Sudoku Math Puzzle

Cosmologicon wrote:It may be possible to brute-force the 9x9 as well, at least part way. There are 5,472,730,538 unique sudoku if you count the symmetries as identical. Some of these symmetries - like swapping a row or column - leave the magnitude of the determinant invariant, but some - like interchanging all of two digits - don't. If you could come up with a way to find the maximum determinant for all symmetric variations of a given sudoku in under, say, a second, that would bring it into the realm of possibility.

It may belong on math; I don't really know. I certainly don't know the answer; I just have a hobby of taking the determinant of each 3x3 matrix, and then taking the determinant of that 3x3 matrix of that. It seems like there should be some clean cut, easy-peasy example...but there probably isn't.

Mouffles
Posts: 60
Joined: Fri Jul 06, 2007 10:02 am UTC
Location: New Zealand

### Re: Sudoku Math Puzzle

Spoiler:
148
726
593

Determinant = 412, this is the maximum for a 3x3 box.

(should this be spoilerised?) yes
In the spirit of taking things too far - the 5x5x5x5x5 Rubik's Cube.

quintopia
Posts: 2906
Joined: Fri Nov 17, 2006 2:53 am UTC
Location: atlanta, ga

### Re: Sudoku Math Puzzle

In order to make the second question more interesting. . .what is the sudoku puzzle with the greatest sum of determinants of 3x3 boxes? Is it possible to get 3708?

Maurog
Posts: 842
Joined: Tue Jul 10, 2007 7:58 am UTC

### Re: Sudoku Math Puzzle

Easy as Pi:
Spoiler:

Code: Select all

`148  726  593726  593  148593  148  726481  267  935267  935  481935  481  267814  672  359672  359  814359  814  672`
Slay the living! Raise the dead! Paint the sky in crimson red!

quintopia
Posts: 2906
Joined: Fri Nov 17, 2006 2:53 am UTC
Location: atlanta, ga

### Re: Sudoku Math Puzzle

+that also has digits 1-9 exactly once on the two diagonals?

Maurog
Posts: 842
Joined: Tue Jul 10, 2007 7:58 am UTC

### Re: Sudoku Math Puzzle

Still kinda obvious...
Spoiler:

Code: Select all

`148  726  593726  593  148593  148  726935  481  267481  267  935267  935  481672  359  814359  814  672814  362  359`
Slay the living! Raise the dead! Paint the sky in crimson red!