288 Fillings, Twelve 4x4 Sudokus Once You Relabel
A brute-force count gives 288 valid 4x4 sudoku fillings, but treat the digits as symbols and only twelve distinct structures remain.
A 4x4 sudoku is four rows, four columns, and four 2x2 boxes, and it brute-forces in roughly half a second. That run returns 288 valid fillings from an empty grid. The digits are just symbols, so the 4! = 24 permutations of 1 to 4 relabel one filling into another; divide 288 by 24 and twelve structurally distinct solutions remain.
The counting argument is the part I can check by hand: rename the first row to 1 2 3 4 and every structure gets exactly one canonical representative. No clever search, just a refusal to count the same grid twenty-four times.
Skip it if you came for the 9x9 answer; that count is 6,670,903,752,021,072,936,960 and takes a different method. What lingers is 288 grids collapsing to twelve once the symbols stop mattering. Write the brute force yourself — half a second is cheap.