How to Solve the Bacteria Grid Puzzle: Invariance Proof Explained (2026 Guide)

The puzzle begins with a single cell at the origin of a grid. If the spots above and to the right are empty, the cell replicates and moves into those new positions. The original spot becomes vacant while the population doubles instantly.

Most people assume this binary expansion makes clearing a specific area trivial. They start with a tiny 1x1 box and see it takes exactly one move. They move to a 2x2 box and find success in only eight moves.

But the difficulty scale is not linear; it is viciously steep. Reaching for a 3x3 box already requires a near-infinite sequence of moves. The astronomical complexity explodes beyond human intuition before you even touch the target.

Growth is often a mask for a trap you cannot escape.

The replication rule feels like progress, but it is actually a cage. Every move requires space, and space is a finite resource on an infinite plane. You are fighting a war against the geometry of the grid itself.

When a system offers infinite choices, you must search for an invariant quantity. This is a value that never changes, regardless of the path you take. It is the only way to navigate the chaos of a replicating swarm.

We assign a specific weight to every single point on the lattice. The origin starts with a weight of one, while each diagonal line has exactly half the weight of the one before it.