Bacteria Grid Puzzle Solution
Summary
A puzzle involving bacteria replication on a grid asks for the minimum moves to clear all 16 lattice points within a 4x4 box. Starting with one cell at the origin, a bacterium can replicate to the spot above and to its right if both are empty, vacating its original position. While a 1x1 box takes one move and a 2x2 box takes eight moves, a 3x3 box appears astronomically difficult, if not impossible, to clear. The core of the problem lies in an invariant: a weighted sum of cells, where each diagonal line has half the weight of the preceding one. This weighted sum remains constant at one throughout any replication process. To clear a box, the sum of weights of cells outside the box must be at least one.
Key takeaway
For puzzle designers or game developers creating grid-based challenges, you should consider using invariant properties to determine the feasibility of solutions. This method can quickly prove impossibility, saving significant development time on unsolvable problems. Before investing in complex pathfinding or state-space search algorithms, verify that a solution theoretically exists by checking for conserved quantities.
Key insights
A weighted sum invariant proves the impossibility of clearing certain grid configurations in a bacteria replication puzzle.
Principles
- Invariants simplify reasoning about complex systems.
- Weighted sums can reveal system constraints.
Method
Assign weights to grid points such that each diagonal line has half the weight of the preceding one, then sum the weights.
In practice
- Use invariants to assess problem solvability.
- Apply weighted sums to analyze state changes.
Topics
- Bacteria Grid Puzzle
- Replication Rules
- Invariant Quantity
- Weighted Sum
- Impossibility Proof
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Editorial summary, takeaway, and curation by AIssential. Original article published by 3Blue1Brown.