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**Fun Game Theory, Guessing a Number With a "Twist"**

**You and I are playing a game. I begin by picking an integer from 1 to 2011 (inclusive). On each turn you try to guess my number. I then tells you whether your guess is too high, too low, or correct. If your guess is not correct, I add or subtracts 1 from my number (always constructing a new number from 1 to 2011). Assuming you play optimally, what is the minimum number of turns you need to guarantee that you will guess my number?**

I know that if I DIDNT change my number each time, it would take 11 turns for you to guess my number. I also know that the optimal strategy is picking a number directly in the middle each time. But the problem is, suppose you chose 1006 first. Then my number could be 1005, and I would tell you "too high" But my new number could in fact be 1006!

Help is MUCH appreciated!