Bacteria Grid Puzzle Solution

A bacteria replication puzzle on a grid asks for the minimum moves to clear all 16 lattice points inside a box, starting from a single cell at the origin. The key insight is finding an invariant: a weighted sum assigned to each diagonal line (each line having half the weight of the previous) that never changes regardless of moves made. Since the total weight of the entire infinite grid sums to only 4, and the 16-point box already accounts for over 3.5 of that weight, the remaining weight outside the box is less than 1 — making it mathematically impossible for bacteria to ever fully escape the box. The puzzle's answer is that clearing the 4×4 box is impossible, and even the 3×3 box cannot be cleared for the same reason.