11. (16 points) Given a three-digit number 111, each operation involves changing two of its digits to the unit digit of the sum of these two digits. For example: $111 \rightarrow 122 \rightarrow 144 \rightarrow 554 \rightarrow 004$ (leading zeros are allowed). If the goal is to transform 111 into 777, what is the minimum number of operations required $\qquad$?
【Solution】Solution: According to the analysis, reverse derivation:
(1) $777 \leftarrow 770 \leftarrow 700 \leftarrow 755 \leftarrow 778 \leftarrow 988 \leftarrow 944 \leftarrow 995 \leftarrow 455 \leftarrow 441 \leftarrow 221 \leftarrow 111$;
(2) $777 \leftarrow 770 \leftarrow 700 \leftarrow 773 \leftarrow 433 \leftarrow 449 \leftarrow 599 \leftarrow 554 \leftarrow 144 \leftarrow 122 \leftarrow 111$,
(3) $777 \leftarrow 770 \leftarrow 700 \leftarrow 755 \leftarrow 778 \leftarrow 988 \leftarrow 999 \leftarrow 990 \leftarrow 900 \leftarrow 955 \leftarrow 996 \leftarrow 366$
$$
\leftarrow 333 \leftarrow 330 \leftarrow 300 \leftarrow 337 \leftarrow 677 \leftarrow 661 \leftarrow 331 \leftarrow 211 \leftarrow 229 \leftarrow 119 \leftarrow 299 \leftarrow 227 \leftarrow
$$
(4) $777 \leftarrow 770 \leftarrow 700 \leftarrow 755 \leftarrow 778 \leftarrow 988 \leftarrow 999 \leftarrow 990 \leftarrow 900 \leftarrow 991$
(5) $777 \leftarrow 770 \leftarrow 700 \leftarrow$

It is easy to know that at least 10 operations are needed.
Therefore, the answer is: 10.