12. As Nest:
(1) $a, b, c, d$ all belong to $\{1,2,3,4\}$;
(2) $a \neq b, b \neq c, c \neq d, d \neq a$;
(3) $a$ is the smallest value among $a, b, c, d$.

Then, the number of different four-digit numbers abcd that can be formed is $\qquad$
12. 28
$\overline{a b c d}$ when it has exactly 2 different digits, it can form $C_{4}^{2}=6$ different numbers.
$\overline{a b c d}$ when it has exactly 3 different digits, it can form $C_{3}^{1} C_{2}^{1} C_{2}^{1}+C_{2}^{1} C_{2}^{1}=12+4=16$ different numbers.
$\overline{a b c d}$ when it has exactly 4 different digits, it can form $P_{3}^{3}=6$ different numbers.
Therefore, the total number of numbers that meet the requirement is $6+16+6=28$.