It is said that in ancient times, Josephus and 31 other people were captured by barbarians, making them stand in a circle, numbered $1,2,3, \cdots, 32$, and then kill number 1, then kill number 3, and thereafter kill every other person, until the last person remaining was set free, and this person was Josephus himself. Could you tell me Josephus's number?
Solution: In the first round, kill $1,3,5, \cdots, 31$, leaving $2,4,6$, $\cdots, 32$; in the second round, kill $2,6,10, \cdots, 30$, leaving 4,8 , $12, \cdots, 32$; in the third round, kill $4,12,20,28$, leaving 8,16 , 24,32 ; in the fourth round, kill 8,24 , leaving 16,32 ; in the fifth round, kill 16 , leaving only 32 , so Josephus's number is 32 .