12. If:
(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 $\overline{a b c d}$ that can be formed is
$\qquad$
12.28.
When $\overline{a b c d}$ has exactly 2 different digits, it can form $C_{4}^{2}=6$ different numbers.
When $\overline{a b c d}$ has exactly 3 different digits, it can form $\mathrm{C}_{3}^{\mathrm{l}} \mathrm{C}_{2}^{\mathrm{d}} \mathrm{C}_{2}^{1}+$ $\mathrm{C}_{2}^{1} \mathrm{C}_{2}^{1}=12+4=16$ different numbers.
When $\overline{a^{2} d d}$ has exactly 4 different digits, it can form $\mathrm{P}_{3}^{3}=6$ different numbers.
Therefore, the total number of numbers that meet the requirements is $6+16+6=28$.