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Originally Posted by SunKneeMarie
Methinks I am not schooled enough in the way of the pulley. I still can't picture this.
And I'm still giving myself credit for my sponge answer.  Sponge-worthy, I say. 
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There is no friction on the pulley, as the monkey climbs, the rope moves down past him. Part of the trick (but no the main part) is that there is no friction on the pulley, it won't stop the rope from moving. Change the monkey to another lifeless weight and you will have it, for the system to be in equilibrium both weights (of the same amount, as stated) would be resting at the same height, like in a balance.
In a balance, both "arms" need to have the same length, in this case the segments of rope. If one arm is longer than the other, the balance would not rest horizontally.