homework and exercises - Equilibrium of a tower of 2-d blocks

In the image below,

• All the blocks are frictionless & identical with side of unit length, height $h$, weight $w$ & center of gravity at their geometric centers.


• The 2 lowest blocks are on solid ground.


• The distance from the corner of each block to the midpoint of the bottom side of the box above it is given (namely $a1,a2,b1,b2$).


• forces $F_{ac}$ and $F_{bc}$ are the resultant reaction forces exerted by blocks $A$ & $B$ on $C$.

I am interested in the behavior of these blocks immediately after setting them in this configuration and releasing them, or more specifically: For what relation between $a1,a2,b1$ & $b2$:

• do $A,B$ and $C$ move?- case 1


• do only $A$ and $C$ move?- case 2


• do only $B$ and $C$ move?- case 3



• is the configuration stable (does not change at all once set under this condition and then left.)?- case 4

For those interested, here's my approach & what (I think) I know already:

In an attempt to find the limiting conditions (the borderline conditions between equilibrium and non equilibrium), I assumed that initially $C$ will tend to be in equilibrium.(I have no rigorous justification for this assumption, just a hunch that "this isn't where the trouble is").Under this condition, $F_{ac}$ & $F_{bc}$ can be calculated, and the moments due to their "equal and opposites" ($F_{ca}$ & $F_{cb}$) about $P1$ and $P2$ can be obtained as: $$ M_{ca}(x,y)=w(.5+a1-a2-x)/(x/y+1)$$ $$ M_{cb}(y,x)=w(.5+b1-b2-y)/(y/x+1) $$ where $x$ & $y$ are the perpendicular distances of the respective forces from the center of $C$.

With some ad-hoc and shaky logic here's what I arrived at:

• 
  

  when $$w\cdot a_2
  



• 
  
  

  when $$w\cdot a_2 =M_{ca}(x1,.5)\quad \& \quad w\cdot b_2 but if $$\boldsymbol{w\cdot a_2 =M_{ca}(x1,.5)\quad \& \quad w\cdot b_2 \geqslant M_{cb}(.5,x1)}$$ the configuration is stable (case-4).

  
  


• 
  

  when $$w\cdot a_2 >M_{ca}(a1,.5)\quad \& \quad w\cdot b_2 but if $$\boldsymbol{w\cdot a_2 >M_{ca}(a1,.5)\quad \& \quad w\cdot b_2 the configuration is stable (case-4).

  
  


• 
  

  The above conditions with $a$ & $b$ exchanged along with their corresponding variables.

  
  

But I have no way to check this or provide a satisfactory argument for these conditions, especially the parts written in bold (I arrived at it by putting various combinations of arguments for $M_{ca}$ & $M_{cb}$ & thinking about what would happen in each case). Is this set of conditions right?. What would be a good approach with a logical progression of steps to solve it?