12. Donald Duck and Mickey Mouse are in a 10,000-meter race. Mickey Mouse's speed is 125 meters per minute, and Donald Duck's speed is 100 meters per minute. Donald Duck has an electronic remote control that can force Mickey Mouse to move backward. By issuing the $n$-th command, Mickey Mouse will move backward at $n \times 10 \%$ of his original speed for one minute, and then continue at his original speed. If Donald Duck wants to win the race, what is the minimum number of times he needs to use the remote control? (20 points)
12. 13

Solution: The time it takes for Mickey Mouse to complete the race is $10000 \div 125=80$ (minutes), and the time it takes for Donald Duck to complete the race is $10000 \div 100=100$ (minutes).
The time wasted by Mickey Mouse when Donald Duck issues the $n$-th instruction is $1+\frac{125 \times n \times 10 \%}{125}=1+0.1 n$.
When $n$ is $1, 2, 3, 4, \cdots, 13$, the time wasted by Mickey Mouse is $1.1+1.2+1.3+1.4+\cdots+2.3=22.1$ (minutes), which is more than 20 minutes. Since Mickey Mouse arrives early by $100-80=20$ minutes, for Donald Duck to win, he must make Mickey Mouse waste more than 20 minutes. Therefore, Donald Duck must issue at least 13 instructions through the remote control to win the race.

Scoring Reference: (1) Correctly providing the answer, 5 points; (2) For each formula, explaining the reasoning, 10 points; (3) Even if the final answer is incorrect, appropriate points should be given for correctly reasoned formulas.