11. (12 points) There is a new type of problem-solving robot that can solve problems, but it has an IQ balance limit. Each time it solves a problem, it subtracts the score of the problem from its IQ balance, consuming an amount of IQ balance equal to the score. When it solves a problem correctly, its IQ balance increases by 1. If its IQ balance is less than the score of the problem it is solving, it will fail to solve the problem. If Xiaopeng uses a problem-solving robot with an initial IQ limit of 25 to solve a set of problems with scores ranging from $1-10$, what is the maximum score he can achieve? $\qquad$
【Solution】Solution: From the analysis, to get the maximum score, we should start selecting from the questions with smaller scores in sequence. With an IQ balance of 25, we need to see how many questions can be selected at most, and then consider which questions can be selected. Therefore, we should consider:
(1) If all 10 questions are answered correctly, the initial IQ balance would need at least $1+2+3+4+5+6+7+8+9+10-1 \times 9=46$ points, which is impossible; $46-25=21$ points are enough to remove the questions worth $10$ and $9$ points.
(2) If the first 8 questions (worth $1--8$ points) are answered correctly, the initial IQ balance would need at least $1+2+3+4+5+6+7+8-1 \times 7=29$ points, which is also impossible; but it is close to 25 points.
(3) If the first 7 questions (worth $1--7$ points) are answered correctly, the initial IQ balance would need at least $1+2+3+4+5+6+7-1 \times 6=22$ points, which is possible; this 22 is 3 points less than the initial IQ limit of 25, so we can use the question worth 10 points to replace the question worth 7 points, thus obtaining the maximum score.
In summary: As long as the questions worth $1, 2, 3, 4, 5, 6, 10$ points are selected, the maximum score of $1+2+3+4+5+6+10=31$ can be obtained.
Therefore: The maximum score that can be obtained is 31.