I’m reasonably sure there are many more puzzles possible than solutions, as it would seem you could choose a very large set of starting positions that would result in the same solution. While I haven’t read anything authoritative on this, it seems self-evident.
I wonder how many there are if you rule out which numbers are in which position. In other words, say you start a puzzle with a 3 in the upper right corner. Now say you have the same puzzle except that there is a 7 in the upper right corner. The whole thing, when solved, is identical except that the sevens and threes have switched places. I’d consider that an identical puzzle.
What I think that the quote from Exapno says is that the 5,472,730,538 number is what you get when you eliminate rotation (shifting the grid 90, 180 and 270 degrees, so the number in the upper left moves to the lower left, etc.), reflection (when the puzzle is flipped left to right or top to bottom), and relabeling (what you ask about, where all 3s are replaced with 7s and all variations of that).
There’s also permutation symmetry on the rows and columns: I could shuffle the first three columns, for instance, or swap the first three rows with the middle three rows, and still have a valid solution. It’s not clear whether the 5.4 billion figure corrects for those or not. But then, there might be some cases where a row permutation is equivalent to a digit relabelling, which makes the counting difficult.
The definition of what is a “new” puzzle changes how many puzzles there are. The 6,670,903,752,021,072,936,960 figure includes all possible variations, including puzzles with the 3s and 7s swapped as two different puzzles.
If you look at certain similar puzzles as logically the “same”, then the number gets lower. For instance, if you just reverse the puzzle left to right, it is logically the same puzzle, just displayed two different way. Similarly, if you replace each 3 with a 7 and vice versa, it is the has same logical solution, with just a different way of displaying it. By eliminating from the count each puzzle that is logically the same as a prior puzzle, you get the much lower figure of 5,472,730,538.
If you’d like to make up a different definition of what is a “new” puzzle, feel free to count under that definition.
And of course, filled grids aren’t necessarily a single “puzzle.” You can create two entirely different logical exercises (which is the “puzzle” part of the puzzle) by changing what the starting grid looks like.
So, given a single example of a complete grid is it possible to calculate the total number of different but solvable without guessing starting grids there are?
For example, given a starting grid there are 81 possible starting grids where 80 of the values are provided and all can be solved logically. There are X possible starting grids where 79 of the values are provided Y of them can be solved logically. And so on.
So if you can calculate the number of feasible starting grids for a single final grid and multiply it by the number of possible final grids then wouldn’t that be the more useful total for the number of “puzzles” possible?
All of which is negated by the fact that if you gave my wife (who does them a lot, I find them boring) a loop of 200 puzzles I doubt she’d ever recognize it.
Here’s a link that explains how the numbers were calculated in a bit more detail. If you’re a little rusty on your group theory, it may not be a quick read.
Heh, I can almost resolve the same puzzle right after I finish solving it and it’d look new to me. This is particularly so for the really hard ones. That’s one of the “charms” about a Sudoku puzzle. Same puzzle, same solution, but the way you manually get to the solution will likely vary every time!
Define “without guessing”. There was a big discussion about that very question just last week, and the definition of “guessing” seemed to vary from person to person, depending on how much depth a person can keep track of. Unless you just mean “puzzles with a unique solution”?