3. Arrange the numbers $1,2,3, \cdots, 2001$ in a clockwise direction on a circular diagram. First, cross out 2, then continue in a clockwise direction, crossing out every second number until only one number remains. The last remaining number is ( ).
A. 1955
B. 2001
C. 81
I. 163
3. $A$

For the general case, consider arranging the numbers $1,2, \cdots, n$ in a circle in a clockwise direction. Let the last remaining number according to the problem's operation be $a_{n}$. When $n$ is an even number $2 m$, $a_{2 m}=2 a_{m}-1$. When $n$ is an odd number $2 m+1$, then $a_{2 m+1}=2 a_{m}+1$. Therefore, $a_{2001}=2 a_{1000}+1, a_{1000}=2 a_{500}-1, a_{500}=2 a_{250}-1, a_{250}=2 a_{125}-1, a_{125}=2 a_{62}+1, a_{62}=2 a_{31}-1, a_{31}=2 a_{15}+1, a_{15}=2 a_{7}+1, a_{7}=2 a_{3}+1$, and $a_{3}=3$, so $a_{2001}=1955$