A Game Theory Puzzle

Question

I encountered this game theory puzzle that I have no idea how to solve.

In , two players take turn to fill in $1,\dots, 9$ in the circles (each number can only be used once). The player who can make one of the edges to have sum equals $20$ wins the game.

I haven't taken any game theory course, but I have read about Zermelo's theorem. As I believe, without strong reasons, that the game should have a winning strategy, I have tried to consider a case where the game ends in a draw and try to deduce backwards that at some point at least one of the players have made some suboptimal moves. And that if they choose to play optimally, it is possible to guarantee a win. However, since there are too many combinations, I have no idea how to show this generally.

Answer 1

Play 5 in a corner. When your opponent plays $a$, play $10-a$ in the symmetrical spot on the same edge.

Answer 2

in this game the person, who puts the first value in circle can always win, but he has to make pair of 10 on each side on which the second user is putting a value.

always remember sum of 20 is never going to be calculated on the side having 5 in circle, and if the 5 is at edge then only sum of 20 can be achieved from its opposite side