Design a strategy that minimizes the expected number of questions you will ask in the fol- lowing 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.
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}?"