#### Design a strategy that minimizes the expected number of questions you will ask in the following game

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# Design a strategy that minimizes the expected number of questions you will ask in the following game. You have a deck of cards that consists of one one, two twos, three threes, and so on up to nine nines for a total of 45 cards. Someone draws a card from the shuffled deck and looks at its value (hiding it from you). The goal is to determine the value of the card through asking a series of questions, each of which must be answerable with "yes" or "no" (such as "Is the card a nine?"). To answer this question, you should express your strategy as a decision tree. You may either explicitly draw the decision tree or describe its construction in sufficient detail so that I could draw it from your description. If you draw the tree instead of describing its construction, give a brief description of how you came up with it. Furthermore, briefly explain why this minimizes the expected number of questions you will ask in this game. You are not required to give a formal proof. Hint: The first question to ask in the optimal decision tree is "Is the card one of {4, 5, 9}?" Equivalently, the question can be "Is the card one of {1, 2, 3, 6, 7, 8}?"

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