Attempted solution: time-out each time he goes into the rooms he's not allowed to go in. Time-out consists of physically holding and restraining child for approximately 2 minutes at a time (since regular time-outs he couldn't care less about).
Result: time-outs do not work for 20-month-old, even though he screams and cries during each one. He continues to open the doors to the off-limit rooms, getting time-out each time, for over an hour. Note to self: child is extremely persistent.
2nd attempted solution: after spending hours searching and reviewing products, Mom finds door knob covers, $3 at Wal-Mart. The $3 are gladly invested.
2nd (and final!) result: the covers work. After much frustration attempting to figure out the new contraptions and throwing toys at the door knobs, 20-month-old gives up.... for now....