224. NumPy Einsum Intro

Einsum is a one-line DSL that expresses any combination of multiplication, summation, and broadcasting using named axis labels. Once you can read it, every paper's pseudocode becomes mechanical to translate.

Implement: def three_einsum_translations(a, b) where a has shape (B, M, K) and b has shape (B, K, N). Return a tuple of three arrays computed via np.einsum:

1. Batched matmul — shape (B, M, N). Pattern: 'bmk,bkn->bmn'.
2. Sum a over its last axis — shape (B, M). Pattern: 'bmk->bm'.
3. Sum b over its middle axis — shape (B, N). Pattern: 'bkn->bn'.

Constraint: All three results must come from np.einsum — no @, no np.matmul, no np.sum.

Reading einsum:

• Letters on the input side label every axis of every input array.
• Letters on the output side say "keep these".
• Repeated input letters that are not on the output side are summed (contracted).
• Letters appearing on both sides are kept (broadcast or batch).

So 'bmk,bkn->bmn' says: a has axes (b, m, k), b has axes (b, k, n); the shared k is summed, b is kept (batched), and m, n are kept. That is exactly batched matmul.