The total number of sudoku grids was estimated and calculated a few years ago - ~6.7*10^21
For completeness and clarity, I might add that it has been estimated that there are ~10^15 different minimal puzzles per individual grid solution. [No danger of running out of puzzles]
The total number of 9*9 latin squares has been calculated ~ 5.5*10^27. {2 out of the 3 constraints}
- Code: Select all
+---+---+---+
|123|456|789|
|234|567|891|
|345|678|912|
+---+---+---+
|456|789|123|
|567|891|234|
|678|912|345|
+---+---+---+
|789|123|456|
|891|234|567|
|912|345|678|
+---+---+---+ An example of a valid latin square [ignore the boxes]
There are I think many many grids which dont quite satisfy the initial row and/or column constraints.
I suspect the minimum number of cells which dont comply is 2
- Code: Select all
+---+---+---+
|175|386|4*2|
|293|417|*58|
|846|295|713|
+---+---+---+
|458|923|176|
|319|674|825|
|627|851|349|
+---+---+---+
|781|532|694|
|534|169|287|
|962|748|531|
+---+---+---+ 2 cells non-complient - or 25/27 constraints complied with.
+---+---+---+
|5**|136|284|
|*18|245|976|
|*24|978|315|
+---+---+---+
|167|382|549|
|293|451|768|
|485|697|132|
+---+---+---+
|841|523|697|
|359|764|821|
|672|819|453|
+---+---+---+ 4 cells in one box non-complient
+---+---+---+
|78*|123|456|
|*9*|456|123|
|*41|*87|*9*|
+---+---+---+
|157|238|*69|
|2*8|619|347|
|369|7*4|582|
+---+---+---+
|41*|365|7**|
|52*|94*|61*|
|63*|*71|9*8|
+---+---+---+ 19 cells non-complient.
These are some of the constrained grids which made the counting of the grids complicated.
I suppose these are the "grid solutions" when you "guess" a couple of wrong proposition clues - and everybody has made one of these grids when they have made a mistake with their pen and paper puzzle solving !.
In the above grid with 2 non complient cells, it is easily transformed into a valid grid by swapping 2 clues
- Code: Select all
+---+---+---+ +---+---+---+
|175|386|4*2| |175|386|492|
|293|417|*58| |293|417|658|
|846|295|713| |846|295|713|
+---+---+---+ +---+---+---+
|458|923|176| |458|923|176|
|319|674|825| |319|674|825|
|627|851|349| |627|851|349|
+---+---+---+ +---+---+---+
|781|532|694| |781|532|964| r7c7 & r7c8 swapped
|534|169|287| |534|169|287|
|962|748|531| |962|748|531|
+---+---+---+ +---+---+---+ valid grid
So there would appear to be a great number of these invalid grids.
C
