There are only twelve 4x4 sudokus - and a cool trick for finding minimal subsets

An elegant way of finding the 12 solutions

I found the 12 solutions by hand. If you label the four numbers in the upper-left big square 1, 2, 3, 4 by symmetry of permutations, then there's a second symmetry you can apply. The second symmetry is that swapping rows 3&4 or swapping columns 3&4 preserves whether a board is a solution. You can use this symmetry to move the "1" in the lower-right big square from wherever it is to row 3, col 3. Once you've written that "1" down it's pretty easy to find the 3 truly distinct solutions:

A              B              C
1 2 3 4        1 2 4 3        1 2 4 3
3 4 2 1        3 4 2 1        3 4 2 1
4 3 1 2        4 3 1 2        2 3 1 4
2 1 4 3        2 1 3 4        4 1 3 2

Not all the solutions are made equal. Most of them (192 out of 288) have 304 minimal puzzles that solve to them, but a decent chunk (96 out of 288) only has 284 minimal puzzle corresponding to them. Why is that?

Truly-distinct-solutions B and C are almost identical, there's just four squares that flip between them. I suspect you can always turn a puzzle for one into a puzzle for the other, and thus they account for the 192 solutions with 304 minimal puzzles, and truly-distinct-solution A accounts for the 96 solutions with 284 minimal puzzles. That explains why the split is 96/192 (i.e. 1/2). The reason that 304>284 is because the puzzle needs to convey a little bit of extra information to disambiguate between B&C and there are multiple ways to do that. I doubt there's a concise reason why the puzzle counts are exactly 304 and 284 (though the second symmetry does explain why they're multiples of 4).