Can you calculate the force of chair legs on floor? I have kinda formalistic question. Eg. we have normal symmetric chair with 4 legs, and mass $M$. Can we calculate forces of chair legs on floor surface $F_1,F_2,F_3,F_4$ with only this data and why not. How does the assumption, that chair is completely rigid or that it is elastic affect the problem. Is such calculation possible with chair with 3 legs?
A little background to this question. I barely remember statement from lectures from my 1st year classical physics, that such solution is possible for chair with 3 legs, but not for chair with 4 legs unless we don't know how the chair deforms. I am looking for formal reason, why do we need to take deformations into account with 4 legs, but not with 3 (although we could use it even for 3 legs).
 A: If the chair has three legs then there are three equations (net sum of forces equals zero, moments about two horizontal axes) in three unknowns, which determine the forces on each leg unambiguously. If we introduce a fourth leg then we have to take account of the stiffness and deflection of the chair to get a solution.
This is the two dimensional analogue of the following one-dimensional situation. If a horizontal beam with a known load is supported at two points then we can determine the force on each support without worrying about the deflection of the beam. But if we add a third support then we have to take the stiffness and deflection of the beam into account.
A: There are two aspects why you can't calculate the forces from the given data.
Geometry
The geometry is not given to sufficient degree:

*

*Are the legs forming a rectangle? Maybe that's what "normal symmetric" wants to express, but real-world symmetric chairs often have their legs arranged in a trapezoid shape instead.

*Where is the center of gravity of M located with respect to the four legs?

Typically, all relevant dimensions should be given, or at least the relative placement of M's center of gravity with respect to the legs' positions.
Solving the force equations
There are four unknown values, but only three independent equations available:

*

*The forces must sum up to the weight of M.

*The total moment around some front-back axis must be zero.

*The total moment around some left-right axis must be zero.

To compute a moment, besides the forces you also need the geometry, meaning the distances from the axis (see above).
Even if the geometry is known, there are still four unknowns out of three equations, meaning that infinitely many solutions can be found.
You mention the concept of deformation. This means that you can compute the deformation that the chair will experience when applying a given set of F1 ... F4 forces. We assume the ground to be perfectly flat and the legs to be perfectly aligned. Then you have an additional equation from the fact that all four legs are supposed to reach the ground, meaning zero deformation, so only force combinations solving that no-deformation constraint can be solutions.
The exact physics and math behind deformation can become quite complex, so you typically can't write down this fourth constraint as a simple concise equation.
