PROBLEM
We have two students who both begin with 1 point, their utility is equal to the number of points they have. Student $1$ has the first turn, he can decide to end the game and both students receive their current points, hence they have combined utility $(1,1)$. The student can also decide to pass on the game to student $2$ but he will lose 1 point while student $2$ receives $2$ extra points, student $2$ has now the same choices as student $1$, to end or to pass on the game. The game ends when both students have 60 points. What is the Subgame Perfect Nash Equilibrium in this game?
MY APPROACH
We derived the following utility formula for student $2$: when $n$ is even, $u_2 = n$, when $n$ is uneven, $u_2 = n + 2$. Hence, when $n = 58$, $u_2 = 58$ and student $1$ has to make a decision. If he decides passes the game on his utility will reduce by $1$ and $u_2 > 60$, and student $2$ will end the game, hence student $1$ will decide to end the game straightaway. Student $2$ knows this the turn beforehand and ends the game himself when he has $u_2 = 59$ in turn 57. Student $1$ knows this and will end in turn 56 and so on. By backtracking we find out that student $1$ will end the game in his first turn and the Subgame Perfect Nash Equilibrium would be that the student who has the turn in turn $i$ will end the game straightaway.
Can anybody please tell me if I did this correct?
This seems correct to me. For further reading, I recommend you the Wikipedia article on Centipede game. Note, that they're a lot more Nash equilibria but your condition of being subgame perfect eliminates all non-credible threats from this.