Having reached the final stage of a game show, you face an endless row of doors labelled $1$, $2$, $3$, ... The game show host has selected a whole dollar amount, and has put this exact amount as a prize behind the door labelled with that dollar figure.
You have absolutely no clue what prize is for grabs, and you have to continue opening doors until you hit upon the prize. But be aware: each door you open costs you a dime ($0.10), payable to the game show host.
If the opening of a door reveals the prize, it's yours and the game ends. If a door being opened reveals an empty room, the door gets closed again, and the game show host secretly moves the prize one door up (from behind a door labeled $n$ to a door labeled $n+1$), or one door down (from a door labeled $n$ to a door labeled $n-1$). Note: following each closing of a door the prize must move, but it's the game show host's sole decision which move to make.
You want to leave this game with a profit, so it is of the essence to find the prize following a limited number of door openings. However, the game show host will do his utmost to reduce the game show cost. And to make things worse: prior to the game and before the game show host decided on the dollar amount for the prize, you have to specify to him the full strategy that you will follow.
What will this strategy be?
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