Question: In the Pick a Number game, why does the strategy of halving the interval work in a random scenario but not in the scenario presented

In the "Pick a Number" game, why does the strategy of halving the interval work in a random scenario but not in the scenario presented in the text?
In the last challenge, what was the strategy of the ultimate winner of Survivor?
The "hot hand" theory in basketball suggests that when a player has made a couple of shots, they are more likely to make the next shot than they would be if they have just missed a couple of shots. Why might the opposite be true?
 In the "Pick a Number" game, why does the strategy of

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