When does a block of ice begin to float in itself? A motivating example:
A friend and I noticed a frozen water bottle melting at work and began to wonder - at what point in the melting process would the remaining ice be floating in the water generated from melting, as opposed to resting on the bottom of the bottle?
The actual question:
Assume a block of ice completely fills an open-top, cube-shaped container with side length L. As the container sits in ambient air, the ice begins to melt. To simplify, we can assume the melting rate results in a constant reduction in all dimensions (i.e. the side length of the cube of at any time is L-x, where x is the current depth of melting). We can also ignore surface tension and anything else that may cause the melted water to "stick" to the ice. At what point in the melting process does the ice float?
My intuition and preliminary math say that it would float immediately, but I'm interested to hear the thoughts of others.
Note that this similar question has been asked, but refers to a case where the ice cube and the container have different initial geometry.
 A: As soon as there is no adhesion between the ice and the walls (and you say to assume that happens "immediately"), any amount of water is sufficient to produce "some" buoyancy. You are saying the ice initially fills the entire container. I am assuming the ice is at 0°C while it fills the container (otherwise, the ice will expand going from whatever initial temperature, say -20°C, to 0°C and the question no longer has an "easy" answer).
If a small volume $dV$ of ice melts, the liquid will have a smaller volume - roughly $0.9~dV$. If this is due to uniform melting on all sides of the ice, and a thickness $dx$ of ice was "shaved off" on all sides, then a liquid film can surround the ice. This liquid film has to rise to at least 0.9 of the height of the ice in order to provide sufficient buoyancy. 
But since melt water was generated at both the top and the bottom, there is plenty of water available to create a "film" of water underneath the ice (thickness 0.9 dx), and still have enough water left (namely, the melt water from the sides) to create a film of water on the sides that produces the pressure needed to lift the ice.
A: In order to float, the depth of water around the ice must be at least 9/10 of the depth of the ice, because the density of ice is 9/10 of that of water. So as soon as a thin layer of water exists around the ice, it will float. 
The difference in Why does not ice float in this case just instantly after melting? is that there is a large gap between the ice and the container. This gap must be filled to 9/10 of the depth of the ice cube before it will float. In your question there is no gap, so the ice floats immediately as soon as some ice has melted.
How thin a layer of water is required? In theory, there is no lower limit. The layer needs to be only a few molecules wide. A cupful of water is enough to float a ship - see Can a ship float in a (big) bathtub?.
Practically, the ice will not float immediately because of bonds formed between the ice and the container. It will not float until these bonds have been broken.   
