Will a platform only supported by a column in the center fall? (Excluding weather) 
My friend told me that without wind, a building with a platform that only has a column supporting it would be balanced and not fall, given that there is no wind. I said that the platform would fall because gravity would be pushing down on the platform, and the column would not be able to support the outer edges of the platform, and it would fall. It's been a long time since I've done anything with physics, so I would love to find out if either of us were correct, or maybe none of us were.
 A: You seem to have used 'minecraft' to draw the diagram.
In an ideal situation such as that game the tower wouldn't fall, even if someone stood at the edge of the platform it doesn't fall (as far as I remember, having seen people play that game).
In the idealised situation you described, if there were no other weight on the platform, it would be balanced and not fall.
Let's see what would happen in reality if there is no cement between the blocks.
If the centre of mass (COM) of any section of the tower falls outside the base (the base that is supporting that section), it would fall.
For example if a person stood on the top, near the edge of the platform and the combined COM of the person and the platform was outside the area where the platform is supported, then it would tip over.
If there was strong cement everywhere apart from the bottom, then consider where the COM of the whole structure (including the person) is and whether it is outside the base.
If there is strong cement including attaching the base to the ground, then it's clearly less likely to fall, but more details would be needed for someone to help decide whether it would fall or not.
A: If the platform is not sufficiently rigid then it will bend downwards at the edges. If it is not sufficiently strong then the stress that this causes at the centre of the platform will cause it to break.
Assuming the platform is sufficiently rigid and strong that this is not a problem, then the next question is whether the platform is fixed to the top of the column in any way. If it is not fixed then it will only stay balanced on the column if it’s centre of mass lies within the area of the top of the column. If its centre of mass is outside of the top of the column (because the platform is not symmetrical, or not centred, or has an unsymmetrical load) then it will fall off the column.
If the platform is fixed to the top of the column then it will stay on the column even if its centre of mass is some distance outside of the column. The maximum distance that its centre of mass can be displaced outside of the column before it falls off will depend on the weight of the platform and the strength of the fixing.
It is the complexity of questions like this and the importance of getting them right when lives are at stake that makes structural engineering a highly regarded and well paid profession.
