Broadcasting rules

The problem, from first principles

02-elementwise-ops's functions all quietly relied on something never actually explained: adding a scalar to an array, or combining two differently-shaped arrays, just worked. That "just works" is called broadcasting, and it isn't magic, it's a precise rule about which shapes are allowed to combine and what shape the result gets. Without knowing the rule exactly, you can't predict when two mismatched shapes will silently combine (possibly computing something you didn't intend) versus when they'll correctly raise an error, and that unpredictability is exactly where subtle shape bugs come from later in this curriculum.

From theory to code

Implement broadcast_shapes(shape_a, shape_b), returning the shape two arrays of those shapes would broadcast to, or None if they're incompatible. Theory's alignment-from-the-right-then-compare-pairwise procedure maps directly onto the function: pad the shorter shape, then walk both shapes together checking one compatibility condition per dimension.

Constraints

• Shapes are aligned starting from the trailing (rightmost) dimension; the shorter shape is conceptually padded with 1s on the left, not the right.
• Two aligned dimensions are compatible if they're equal, or if either one is exactly 1.
• The result dimension for a compatible pair is max(dim_a, dim_b).
• If any aligned pair is incompatible, return None immediately — there's no such thing as a partial broadcast.
• Must handle shapes of different lengths (including a () scalar shape) and shapes of equal length.

Hints

Open one at a time. Each gives away a little more than the last.