Why does an ice-cube turn faster and faster while melting in water? 
Whenever I put an ice-cube into a glass of hot water, so that it melts quickly, and it is initially rotating slowly, I noticed that its rotational speed increases as it melts and 'shrinks'. Why?

I thought of conservation of angular momentum, but does that really apply here? The part of the ice that melts into water would be 'carrying' off some of the ice-cube's angular momentum as it breaks away and diffuses into the surrounding water. So I see no reason for the remaining ice-cube to rotate faster.
I can only think of one possible reason, and it is just a guess: The friction between the water and ice causes the melting part to form eddies that spin in the opposite direction to the ice-cube, and so the ice-cube loses much less momentum in melting since its boundary is actually moving together with the adjacent eddy boundaries. If the eddies expand as the ice-cube shrinks, it seems we would have to conclude that the ice-cube must rotate faster to preserve total angular momentum.
But such an analysis seems very weird, and cannot be a proper picture of what is really happening since the 'number' of eddies must decrease as the ice-cube shrinks, so they must combine in some manner, and I have no idea how any of this works out.
So, what really is spinning the ice?
 A: The same physics that explains why water tends to rotate when you drain your bathtub also answers this question.  In both situations there is a downward flow of water that accentuates the residual rotation that remains after the vessel has been filled.  In the case of your bathtub the downward flow results from opening the drain.  For the ice cube in hot water the downward flow is created when the just melted water sinks because it is colder than the surrounding water.  This downward flow creates a vortex and it is the interaction between the ice cube and the vortex that causes the increase in rotational speed ot the ice cube.
There is an urban legend that claims that the direction of spin in drains is associated with the coriolis effect and differs between the northern and southern hemispheres.  While this may be true of hurricanes and cyclones, it is too small an effect to apply to bathtub drains or glasses of hot water with ice cubes.  For these smaller scale systems the rotation direction is dictated by the residual rotation created when the vessel was filled.  This small residual rotation may not be noticeable until the downward flow accentuates it.
Edit: @sammygerbil has found a website that discusses this phenomenon and attributes it to the same mechanism I outlined above.
A: This is a very nice question, to start with! Let me turn to what is already described in the website sent in by sammygerbil. When the rotating ice cube melts, the cold melting water is moving downwards under the cube, taking away angular momentum from it. The vortex beneath the rotating cube gets less when approaching the bottom because the rotational momentum is taken up by the hot water surrounding it.
The downward moving cold water attracts hot water (the cold water moving downwards must be filled up by other water and this can be both hot water or melting water, but the hot water prevails, as might be clear) which gains linear momentum, and this linear momentum is subsequently conversed (by a torque acting on the cube caused by the attracted hot water) in an increased angular momentum of the ice cube.
So I don't think the cube gets to spin faster because a smaller cube is easier to rotate in the whirlpool, because the whirlpool beneath the melting cube also gets smaller while melting (scale invariance).
