The New Sudoku Players' Forum

[Maq777]

Knowing that the first box can be filled in 9! (362.880) different ways, and after choosing one of them, the second box can be filled in 12.096 ways, and once boxes 1 and 2 are filled, the third box only admits 216 forms to be filled.

My question is, given a case like this

Code: Select all

 1 2 3 | 9 7 8 | . . .
 4 5 6 | 3 1 2 | . . .
 7 8 9 | 6 4 5 | . . .
 - - - | - - - | - - -
 9 7 8 | 1 2 3 | . . .
 3 1 2 | 4 5 6 | . . .
 6 4 5 | 7 8 9 | . . .
 - - - | - - - | - - -
 . . . | . . . | ? ? ?
 . . . | . . . | ? ? ?
 . . . | . . . | ? ? ?


How many different ways can box number 9 be filled?

What kind of algorithm or mathematical interpretation do we have to find the exact value?

Best regards

[JPF]

There are 9!=362880 ways to fill the Box 9
8784 give puzzles having 64 solutions
Here is an example:

Code: Select all

+---+---+---+
|123|978|...|
|456|312|...|
|789|645|...|
+---+---+---+
|978|123|...|
|312|456|...|
|645|789|...|
+---+---+---+
|...|...|123|
|...|...|456|
|...|...|789|
+---+---+---+    64 solutions


354096 give invalid puzzles (no solution)
example:

Code: Select all

+---+---+---+
|123|978|...|
|456|312|...|
|789|645|...|
+---+---+---+
|978|123|...|
|312|456|...|
|645|789|...|
+---+---+---+
|...|...|123|
|...|...|457|
|...|...|689|
+---+---+---+    no solution


Brute force with gridchecker.

more:
depending on the boxes B1B2B4B5 and B9 the number of solutions is 0 or 2^k (k=0,1,2,...,6)
see this post from 2006!

JPF

[Maq777]

Excellent.

Thanks a lot