6.34 In a certain chess competition, there are two seventh-grade students and some eighth-grade students participating. Every two participants play one game, with the winner getting 1 point and the loser getting 0 points. If it is a draw, each gets half a point. It is known that the two seventh-grade students scored a total of 8 points, and all eighth-grade students scored the same number of points. Find the number of eighth-grade students participating in the chess competition.
[Solution] Clearly, the scores of the participants are all integers or half-integers (i.e., multiples of 0.5). The average score of the eighth-grade students among themselves is, of course, an integer or a half-integer, so the average score they get from the matches with the two seventh-grade students should also be an integer or a half-integer.

The two seventh-grade students scored a total of 8 points, and the match between them scored 1 point, so they scored a total of 7 points in the matches with the eighth-grade students.

Assume there are $n$ eighth-grade students participating, so they scored a total of $2 n-7$ points in the matches with the two seventh-grade students, with an average of
$$
\sigma=\frac{2 n-7}{n}=2-\frac{7}{n} \text {. }
$$

To make $\sigma$ an integer or a half-integer, $\frac{7}{n}$ should be an integer or a half-integer, which is true if and only if $n=7$ or 14. When $n=7$, each eighth-grade student scores an average of 4 points; when $n=14$, each eighth-grade student scores an average of 8 points. It is easy to provide examples showing that both scenarios can exist, so the number of eighth-grade students participating is 7 or 14.