10. (18 points) In the figure with 9 circles connected by 10 straight lines, each line has 3 circles. Player A and Player B take turns choosing an unselected circle; if a player selects 3 circles that lie on the same line, that player wins. Now, Player A has chosen “1”, and Player B can then choose “5”. For Player A to win, the product of the numbers of the next possible choices is

保留源文本的换行和格式，翻译结果如下：

10. (18 points) In the figure with 9 circles connected by 10 straight lines, each line has 3 circles. Player A and Player B take turns choosing an unselected circle; if a player selects 3 circles that lie on the same line, that player wins. Now, Player A has chosen “1”, and Player B can then choose “5”. For Player A to win, the product of the numbers of the next possible choices is
504
【Solution】Solution: According to the problem, we have:
(1) If 2 is chosen, then Yi must choose 3, Jia must choose 7, Yi must choose 4, Jia must choose 6, Yi must choose 9, and Jia cannot win.
(2) If 3 is chosen, then Yi chooses 2, Jia chooses 8, no matter what Yi chooses, Jia wins.
(3) If 4 is chosen, Yi chooses 8, Jia chooses 2, no matter what Yi chooses, Jia wins.
(4) If 6 is chosen, the sequence of Jia and Yi taking turns is 6324789 or 6284739, and Jia can win in both cases.
(5) If 7 is chosen, then Yi chooses 4, Jia must choose 6, Yi then chooses 8, Jia wins by choosing 2; if Yi chooses 2, Jia wins by choosing 8; if Yi chooses 3, Jia wins by choosing 9; if Yi chooses 9, Jia wins by choosing 3; if Yi chooses 8 or 2, Jia wins by choosing 2 or 8. If Yi chooses 3 or 9, Jia wins by choosing 4; if Yi chooses 6, Jia wins by choosing 4.
(6) If 8 is chosen, Yi must choose 4, Jia must choose 6, Yi must choose 3, Jia chooses 7, Yi chooses 9, and Jia cannot win.
(7) If 9 is chosen, Yi chooses 2 or 8, Jia chooses the opposite 8 or 2, Jia wins; if Yi chooses 3, Jia chooses 7, Yi chooses 8, Yi wins. Therefore: $3 \times 4 \times 6 \times 7=504$.

Therefore, the answer is: 504.