Centre of mass of every sub-stack against its support, margin per level, and the H(n)/2 overhang curve.
Bisects the scale of the harmonic stack for n = 2..12 until the static margins vanish and compares the overhang with H(n)/2; also finds the smallest n for overhangs of 1, 2 and 3 block lengths (theory 4, 31, 227).
Not run yet.
The science
Statics rule. A stack of rigid blocks stands if, for every level, the centre of mass of everything above it lies over the support below. Put the top k blocks' centre of mass exactly over the edge of block k+1 and block k sits l/(2k) beyond block k+1; the total overhang is d(n) = (l/2) Σk=1..n 1/k = (l/2) H(n) ≈ (l/2)(ln n + 0.5772). The harmonic series diverges, but very slowly: one block length needs 4 blocks, two lengths 31, three lengths 227.
What this 2D view shows. For every level the dot is the combined centre of mass of the blocks above it, and the bar under the level is the contact interval it must stay inside. The left graph is the margin of every level in block lengths (negative = unstable); the right graph is H(n)/2 against n with the 1-block and 2-block thresholds. In the uniform shift mode every block is shifted by the same step; the limit is then about l/n per block, so the overhang grows only like a constant, not like ln n.
Toppling. When a level is unstable the sub-stack above it rotates as a rigid body about the contact edge: Ip φ'' = m g (xc cosφ + yc sinφ) with xc, yc the centre of mass relative to the pivot and Ip the composite moment of inertia about it (parallel-axis theorem). Smaller gravity (Moon, Mars) only slows the fall by √(gEarth/g); it does not change which stacks stand.
Honest simplifications. Identical, perfectly rigid, homogeneous blocks (l = 0.20 m, thickness 0.04 m); frictionless-contact-free statics (friction is assumed sufficient not to slide; the 3D lab includes a Coulomb-friction rigid-body solver and a tilting table); a single critical level topples at a time; after rotating past 90° the animation stops. Real blocks give a little less than the theoretical overhang because of edge crushing and imperfect placement. The harmonic stacking is the classic one-sided brick-stacking result (Paterson et al. 2009 show that counterbalanced, non-one-sided stacks can reach order n1/3 block lengths).