If we repeatedly divide a solid in half, at what point does it stop being a solid? Suppose I have some material in solid-state (say), I cut it into two parts. Take the first cut it into two parts, take the first cut it into two parts, and then repeat this again and again. There will be a point when the substance loses its solid property. I'm interested in this point.
I realize we don't have to go until we break it into two molecules but the state will come a little sooner. This thought experiment is a little bit crazy (and at a point impossible) but please consider this and correct me if there is a flaw in my thinking.
 A: Generally speaking, there isn't a hard line: the borders between regimes are fuzzy, and their positions can be quite different depending on what properties you're looking at.
Moreover, it is generally rare to have a direct border between "molecule" behaviour and "solid" behaviour: the intermediate size regimes typically behave very differently to both of those extremes, and they need to be handled separately. Depending on their size, these materials are known as atomic clusters or nanoparticles (though several other related, more technical terms are also important). Both of those regimes are the focus of active, dedicated fields of current cutting-edge research.
A: It depends on what you mean by 'losing its solid property'. If you say that means it has different properties than maybe nanoparticles would be a good cutoff. A nanoparticle ranges from about 1 - 100 nanometer . Atoms are usually between 0.1 - 0.5 nanometer. Nanoparticles can have vastly different properties from bulk solids because the surface layer is a substantial part of the particle. Note that the surface layer extends a couple atoms into the solid. For example gold nanoparticles can have many different colours (see picture).
The fact that the size of the particle changes the properties of the material is not confined to nanoparticles. While the effects are strong in the nanoregime this also happens for microparticles (1 - 1000 micrometers) and any other scale. So it depends on the material when its properties will change. A gas for example has very little interaction between its molecules so you will generally see little change as you cut it up.

Image source: https://www.sigmaaldrich.com/technical-documents/articles/materials-science/nanomaterials/gold-nanoparticles.html
A: 
There will be a point when the substance loses its solid property.

This sounds like a version of the Sorites Paradox in philosophy: If you keep removing grains of sand from a heap of sand, at what point is it no longer "a heap"?  Or if removing 1 grain can't ever be the difference between a heap and a non-heap, doesn't that imply that 1 grain of sand is still a heap if you repeat the process until then?
There's only a problem if you insist on binary logic, which by definition has hard cutoffs and no fuzzy boundary, leading to non-intuitive results for qualitative descriptions of quantity such as the English word "heap". Fuzzy logic (this small collection of grains "is a heap" is only 0.25 true, 0.75 false) is closer to how humans think, and is (IMO) the most sensible way to resolve the paradox.
Applied to your question, we see that you've created a false premise of there being a sharp cutoff point.
In reality, assumptions (and formulae) that work for large solid objects work less and less well on smaller objects, with probably some aspects "stopping working" sooner than others.
e.g. I think momentum and kinetic energy continue to work until your object is so small that Heisenberg uncertainty is significant.  For kinetic energy, maybe when it's hard to make a distinction between kinetic energy vs. thermal energy.  But I'm not an expert in this area of physics.  @Kai commented under another answer that https://en.wikipedia.org/wiki/Mesoscopic_physics is a separate subject between solid-state and particle physics.
A: Suppose you start in the thermodynamical equilibrium of the solid at a given temperature and suppose "cutting" means also moving apart from each other by the mean free path in the gas-phase. For doing this you have to apply work. If you are finished with cutting (say you did this in the order of 3*Avogadro number of times) you end up with a non-equilibrium gaseous state, to do so you had to invest an amount of work equal to the heat of sublimation, but the temperature of the surrounding heat bath will still be a temperature where the substance is solid in the TD equilibrium, and that will relax pretty fast back to the solid state under release of energy (heat).
