Work backwards. You want a finished 12" block. You're sewing 4 equal size patches together to make it. That means each of those patches needs to be 6" finished, which means they are 6.5" unfinished. Now if those 4 patches were part of one big block (the 9-patch) then the 9-patch would have been 13" unfinished, or 12.5" finished. That means each patch in the 9-patch is 12.5"/3 finished = 4 1/6" (4.166666") finished, or 4 2/3" (4.666666") unfinished. Your strips need to be 4 2/3" wide.
Now to check the answer: Take 4 2/3" wide patches, sew into a 9-patch, you lose 1" with two seams so you have an unfinished block that is 3*(4 2/3) - 1" = 13". Cut into 4 pieces to get 4 patches that are 6.5" unfinished, this time you lose only 1/2" when you sew them together, so the resulting D9P (unfinished) is (2*6.5) -.5 = 12.5"