12. (6 points) There are 2013 students participating in a math competition, with a total of 20 competition questions. Each student has a base score of 25 points. Additionally, answering a question correctly earns 3 points, not answering a question earns 1 point, and answering a question incorrectly deducts 1 point. Then the total score of all participating students is (fill in “odd” or “even”).

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12. (6 points) There are 2013 students participating in a math competition, with a total of 20 competition questions. Each student has a base score of 25 points. Additionally, answering a question correctly earns 3 points, not answering a question earns 1 point, and answering a question incorrectly deducts 1 point. Then the total score of all participating students is (fill in “odd” or “even”).
12. (6 points) There are 2013 students participating in a math competition, with a total of 20 competition questions. Each student has a base score of 25 points. Additionally, answering a question correctly earns 3 points, not answering a question earns 1 point, and answering a question incorrectly deducts 1 point. Then the total score of all participating students is (fill in "odd" or "even").

【Solution】Solution: Each person answers $x$ questions correctly, does not answer $y$ questions, and answers $z$ questions incorrectly, so obviously $x+y+z=20, z=20-x-$ $y ;$
Therefore, a student's score is:
$$
\begin{array}{l}
25+3 x+y-z, \\
=25+3 x+y-(20-x-y), \\
=5+4 x+2 y ;
\end{array}
$$
$4 x+2 y$ is clearly an even number, and $5+4 x+2 y$ is definitely an odd number;
The sum of 2013 odd numbers is still an odd number.
Therefore, the total score of all participating students is an odd number.
So the answer is: odd.