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I did all the puzzles and reached "The Last One" which I am guessing is not actually possible to solve? Unless I wasn't even supposed to try it and instead go somewhere else. Wasn't going to play it again to see if there was another option though as finding all of the puzzles was annoying (especially without mouse-look).

(+1)

Yes it is impossible. It is slight adaptation (as I cannot use two edges between two vertices) of classic riddle/problem Seven Bridges of Königsberg.