OMG, your response just made me realise that the box could be used to change MY cycle. I was doing super complicated manipulations to change the enemy's cycle one by one, by pushing the box where they would have moved. This litterally made my time on the 11th level, like, 10 or 20 times shorter! I suddenly feel a whole lot of not smart right now XD
So in conclusion, good game, and I WAS doing something very weird.