A block tower drawn as an elevation and a top view: where the centre of mass of everything above sits on every layer, and how much support margin is left.
The science
A rigid body at rest on a support is stable if the vertical line through its centre of mass falls inside the support polygon, the convex hull of the contact area. For the tower, layer k rests on layer k-1. Blocks are 1 x 1 x 3 units, three per layer, with neighbouring layers crossing at right angles. The contact area is the set of unit squares where a block of layer k sits on a block of layer k-1; its convex hull is the support polygon. The load is everything from layer k upward (plus an optional cap weight): r_cm = sum(m_i r_i) / sum(m_i).
The margin of a layer is the distance from the centre of mass to the nearest edge of the hull, in block widths; it is negative when the tower would topple at that level. Removing a block narrows the hull of the layers around it. Taking out both outer blocks leaves a narrow strip with a small margin, taking out the middle one leaves a wide hull and a large margin, which is why the middle block is the safe pull in the game.
Simplifications: a static test only (no friction, nudge dynamics or rotation of blocks, which the 3D page simulates), identical blocks, a perfectly aligned tower. Check below: the same support-hull logic reproduces the harmonic overhang (1/2)(1 + 1/2 + ... + 1/n) of a stack of bricks.