Suppose that you have a stack of identical, frictionless, uniform-density $1 \times 2$ bricks arranged in the ordinary configuration (each row is offset by $1$ relative to the row below.)
Is there a combinatorial rule that captures whether or not a given stack of bricks would be stable? In particular, that no small vertical force will cause the stack to move.
In other words, how can one determine whether or not a configuration is stable without doing a static force analysis.
(Admittedly, this question is a little hand-wavey, so let me know if I can clarify anything.)
Clearly, a stack of bricks that does not have any "overhangs" should be stable:
But we can also allow a cantilever if there's a brick above:
However, having a brick above is not sufficient, because a stack like this should not be stable.
Similarly, having a brick above is necessary because a configuration like this is in unstable equilibrium; in particular the upper-right brick in this example would fall if an arbitrarily small force were applied to the right side.