This is really about entropy. Hot things are full of random motion, and cold things are relatively still. It is highly improbable for a system to maintain one region in which there is far less motion than in another region, because the movement of particles causes them to space their motion out. So the ice cubes and the liquid tend towards the same temperature, at the average of their heats. No energy has been "sucked" away, nor has anything "cooled" the other thing - purely on the basis of conservation of energy, the ice cubes could get colder and the liquid hotter in a way that conserved energy. It's entropy that governs the movement of systems to their average heat.
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