Dear forum:

I am a recently retired engineer, and I've been using some of my new-found idle time writing a sudoku solving program. It should be straightforward, and will allow me to find further methods for manual solving.

I've read many of the #puzzles entries. I've probably missed it in my haste, but I found little about the many obvious symmetries in sudoku.

In any puzzle a 1 --> 8 constant may be added, mod 9, to every cell, resulting in perfectly valid unique versions of the puzzle. In the postings I see every cell being varied. Why not count the puzzles with (row 0 col 0) = 1, with perfect confidence that the total number will be precisely nine times that value?

Taking it one step further, fix the value of R0C2 = 1. May we not assume that the total number will be 9 * 8 = 72 times the total we calculate?

My greatest strength as an engineer was the highly successful use of my intuition in developing complex algorithms. And of course some of my worst mistakes resulted from relying too much on that same intuition. So, let me be the hero or the goat...

How far will symmetry allow us to go? If we arbitrarily set the upper left cells of every 3x3 block to the numbers 1 --> 9

R0C0 = 1 R0C3 = 2 R0C6 = 3

R3C0 = 4 R3C3 = 5 R3C6 = 6

R6C0 = 7 R6C3 = 8 R6C6 = 9

will the number of valid sudoku puzzles be precisely 9 * 9! = 3,265,920 times the number of valid puzzles with the nine given cell values? If my suppositions are correct this should save quite a few PC minutes.

One more suggestion. Every valid sudoku puzzle becomes another valid puzzle if it is flipped horizontally or vertically. I don't have the programming or mathematical expertise of many of the other participants, but it should be feasible to quickly identify such reflections, reducing the analysis by another factor of precisely four.