It seems there is no limit of the sufficient days to find the bum.
Since the police can always only search one cellar per day but the bum has two choices to go next day, even the police can deduce which cellar the bum was in last day, they cannot make sure which cellar the bum must be in the next day. Therefore, there is always possiblity that the bum can appear in a cellar where the police is not searching.
Since the police can always only search one cellar per day but the bum has two choices to go next day, even the police can deduce which cellar the bum was in last day, they cannot make sure which cellar the bum must be in the next day. Therefore, there is always possiblity that the bum can appear in a cellar where the police is not searching.
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