The concentration of water outside potatoes B and C is more than the concentration of water inside the potatoes. So water will flow inside the potato. Now, since the concentration of water in the hollowed portions is less, water will move inside the hollowed portion and hence fill the hollowed portion of potatoes B and C. I think since potato D is boiled and hence it's cells are dead it can't perform Osmosis and water does not fill inside the hollowed portion of potato D. Is my reasoning correct ? And I can't get why there won't be any water in hollowed portion of potato A. Is it that the hollowed portion of potato A does not contain any solution so no water would go there and if this reasoning is correct then it implies that osmosis is only possible when there are two solutions with different concentrations and not in any other case ?
Osmosis is actually taking place between the sugar/ salt solution and the cytoplasm inside the potato cells. Or if you want to expand more on this, then water is flowing first from the water in the trough to inside the cells, and then from the cells into the solution present in the hollow of the potato
Osmosis occurs in cup b and c because the sugar or salt makes a concentrated solution which is hypertonic to the cytoplasm, hence water flows out through partially permeable cell membrane by osmosis.
Cup a is an experimental control. It contains no sugar/salt so it can be proven that it was the concentration difference between the solutions that was causing osmosis. No sugar means no solution, just nothing over there(no water , no sugar, no salt, no solution) so osmosis can't occur.
As far as D is concerned, you are wrong that osmosis requires being alive. Osmosis is a natural passive/automatic process, it can happen in dead things too. It's hard to say what can be causing the osmosis to stop. The best guess would be that boiling damages the cell membranes of the potatoes so that aquaporins(water channels in the cell membrane through which water flows naturally by osmosis) are destroyed. Thus the water can't flow through the damages cell membrane.