## Constrained Grid Solutions

Everything about Sudoku that doesn't fit in one of the other sections

### Constrained Grid Solutions

A sudoku solution grid is a 9*9 latin square with an additional box constraint. {3 [row/column/box]equivalent constraints}

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
coloin

Posts: 1915
Joined: 05 May 2005