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I am designing a treasure hunt game where users start of at a fixed number of steps (x) away from the treasure. The user responses are either a or b. A correct answer places the user one step closer to the treasure, a wrong answer takes the user one step away from the treasure. (maximum steps away from the treasure is x.). The two options have a 50% chance of being right on each question. The Game ends when player is 0 steps from the treasure.

Is it correct that the user will always get to the treasure in 2x moves? Or is it 2^x (2 power x)?

Is there anything a player can do to increase their chances of winning? Is there a way I can make the game harder. (Apart from increasing x)?

Please keep the answer simple.

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    Have you studied stochastic processes? This would be an exercise if you had. I could explain it to you if you did, but if you didn't study those, I can give you the answer plain.2011-08-08
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    I have seen the phrase stochastic processes for the first time ever in your post. I am now on Google to learn more about it/them.2011-08-08
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    It'll take a while before you can find the answer just by Googling... but it's a good thing you do =) I'll find the answer and post it for others to see... except if someone does before me =)2011-08-08
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    You are dealing with the most studied example of a **random walk**. Lots of results. One can get a recurrence, and then an explicit formula, for being in ant "state" at a given time.2011-08-08
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    A player cannot do anything to increase his chances of winning (yelling at your console hoping to get lucky won't work =D) but if you want to make the game "harder" (i.e. make the player win less quickly) you could make your random walk assymetric, say, there is, for instance 3 choices, $a$, $b$ and $c$, and a correct answer (say $a$) would make the user move forward, and both $b$ and $c$ would make you move backwards. Another way you could do is let $a$ make you move forward, $b$ you stay in place, $c$ backwards. It will take more steps to get to $x$.2011-08-08

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