I saw 2 people playing a board game variation of this and immediately wondered what the best strategy for this game is:
Consider a path of length n (Each player's token is on spot 0 and the first player to move up to spot n wins). Player 1 chooses a (integer) number between 1 and 3. Player 2 guesses which number was chosen. If player 2 gets the number incorrect player 1 moves his token up the amount of spaces his number was (e.g. Player 1 chooses 2, Player 2 says 3, Player 1 moves up 2 spaces). Player 1 then chooses another (could be the same) number and repeats the process. If player 2 gets the number correct Player 1 doesn't move. Player 2 gets a turn to choose the number while Player 1 guesses and play countinues this way, alternating everytime a player guesses the number correctly.
My question is: What is the best mixed strategy for this game (of arbitrary length n)?
(Note that overshooting the goal counts as a win too.)