No of sudokus, A107739

[Shripad M. Garge]

Indeed the condition of having the boxes in a sudoku to be squares is
natural enough and the natural solution might be to make two sequences. 

The sequence a(k) will count the number of sudokus in a k^2 X k^2 grid 
with k X k squares in it. 

Another sequence b(k) will count the number of all possible sudokus with
m X n boxes in the k X k grid where m.n = k (possibly modulo the 
equivalence that sudokus S_1 and S_2 are equivalent if one can be obtained
from the other by one of the 8 symmetries of the k X k grid, four
rotations and four reflectins). 

My remark about b(p) = 1 was of course wrong. 

Regards,
Shripad. 

:: In a sudoku, rows, columns, and boxes must be filled using the same set of 
:: numbers, which implies that rows, columns and boxes will all have the same 
:: number of cells.  An immediate consequence is that a sudoku is square.
:: 
:: If, in addition, we require that the box be square, then it will have a 
:: square number of cells, implying that the rows and columns will as well. 
:: This means that the sudoku will have side n = k^2 (k >= 1).  This gives a 
:: natural way to write sudoku sequences, with a(k) referring to sudokus of 
:: side k^2.
:: 
:: If we allow a box to be rectangular, there will d(n) sudoku formats of side 
:: n, where d(x) is the number of divisors of x.  For example, a 6 x 6 sudoku 
:: might have 1 x 6, 2 x 3, 3 x 2, or 6 x 1 boxes, giving d(6) = 4 sudoku 
:: formats.  If we consider an a x b box equivalent to a b x a box, there are 
:: ceil(d(n)/2) sudoku formats of side n.  If we further require the boxes to 
:: be distinct from rows or columns, we are left with ceil(d(n)/2)-1 distinct 
:: sudoku formats of side n.
:: 
:: These conditions imply that for primes p, q:
:: 
::     - if n = 1 or n = p, there are no sudoku formats of side n.
::     - if n = pq or n = p^3, there is a unique sudoku format of side n.
::     - otherwise, there are more than one sudoku formats of side n.
:: 
:: For example, a sudoku of side 7 has no acceptable boxes.  The only 
:: possiblity, a 1 x 7 box, is equivalent to a row.  A sudoku of side 10 could 
:: have a 2 x 5 box, this is the only possibility.  A sudoku of side 12 could 
:: have a 2 x 6 or 3 x 4 box.
:: 
:: The upshot is, if we allow rectangular boxes, there can be any number of 
:: soduko formats of a given side.  This makes a sequence of statistics on such 
:: sudokus somewhat complicated.  What would a(12) count, the number of sudokus 
:: with boxes of size 2 x 6 or of 3 x 4, or both?
::