The initial state of every lock = locked
On every NON-SQUARE numbered door, the number of wanders changing the state will be EVEN. And hence the door will return to locked position.
E.g. Door No.$8$: Warders $1, 2, 4$ and $8$ will act.
Door No.$30$: Warders $1,2,3,5,6,10,15,30$ i.e. a total of $8$ warders will act.
On every SQUARE numbered door, the number of wanders changing the state will be ODD. And hence the door will be in unlocked position.
E.g. Door No.$16$: Warders $1, 2, 4, 8$ and $16$ will act.
Door No.$36$: Warders $1,2,3,4,6,9,12,18,36$ i.e. a total of $9$ warders will act.
So door No. $\color{gold}{1, 4, 9, 16, 25, 36, 49, 64, 84}$ and $\color{gold}{100}$ will be in Open position and rest in Locked position.
Concept used: Perfect squares have odd number of factors and others have even number of factors.