If we write every day of 2014 in the form of an eight-digit number, for example, 20140125 represents January 25, 2014, how many eight-digit numbers have the digits '1', '2', '0' appearing the same number of times?
Counting.
Let every day of 2014 be written in the form of a number $\overline{2014 A B C D}$;
where, the first four digits '1', '2', '0' appear the same number of times; then the last four digits '1', '2', '0' also appear the same number of times.

When $\overline{A B}=01$ or 10, $\overline{C D}=23 \sim 29$, there are $2 \times 7=14$ cases;
When $\overline{A B}=02$, $\overline{C D}=13 \sim 19$, there are 7 cases;
When $\overline{A B}=03 \sim 09$, $\overline{C D}=12$ or 21, there are $7 \times 2=14$ cases;
When $\overline{A B}=11$, it does not exist;
When $\overline{A B}=12$, $\overline{C D}=03 \sim 09$ or 30, there are 8 cases;
In summary, there are $14+7+14+8=43$ eight-digit numbers where the digits '1', '2', '0' appear the same number of times.