Guessing Game With Optimal Profit

Discrete Mathematics Level 4

One day, you decide to play a game with your friend out of sheer boredom. Your friend picks out an integer randomly from the range \([1,1023]\) and you try to guess it.

You can guess an infinite number of times. Let \(n\) denote the integer your friend chose and let \(k_i\) denote the integer you chose on the \(i^\text{th}\) try. If \(n > k_i\) your friend will tell you "higher", if \(n < k_i\) your friend will tell you "lower" and if \(n = k_i\) then you win the game.

But then, your friend decides this is too boring. He decides to give you 1023 dollars. Every time you make a guess, right or wrong you have to give him \(x\) dollars, even if you already gave him back the original 1023 dollars or even if you already have negative profits. Since your friend knows that you will play optimally, he sets \(x\) such that your expected profit after playing the game is 0. Find \(\lfloor x \rfloor\).

Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.


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