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When the walls are true and vertical - they at least stand a chance!!
I take it you mean the area of the floor is 20 sq metres? In which case the ceiling would be the same size, but you can't calculate the area of the walls without knowing their height.
No. if the floor is smooth you must have friction between the ladder and the floor, or the ladder would just slide away. Technically you could do without friction between wall and ladder, but even that can be helpful in preventing the ladder from going sideways.
(21.5) x (21.5) = 462.25 -- It has 462.25 square feet on the floor. -- It has another 462.25 square feet on the ceiling. -- It has (21.5 times the height of the room) square feet on each of the four walls.
It takes 20 seconds for each floor. Therefore, it will take 140 seconds for you to reach the eighth floor.