Bacteria Grid Puzzle Solution

The video analyzes a bacteria replication puzzle where cells start at the origin and can replicate by populating spots above and to the right when both are empty, leaving the original spot unoccupied. The goal was to find the minimum moves to clear all 16 lattice points in a specific box. Starting with smaller cases, a 1x1 box takes one move and a 2x2 box takes eight moves, but attempting a 3x3 box reveals the astronomical difficulty. The key insight comes from finding an invariant quantity - a weighting system where each diagonal line has half the weight of the previous one, with the origin having weight 1. Since each replication converts one cell into two on the next diagonal, the total weighted sum always remains constant at 1. By calculating the sum of all weights on the infinite grid (which converges to 4) and comparing it to the weights inside the target box (3.5+ for the 16-point box), the analysis shows that the remaining weight outside the box is less than 1, making escape mathematically impossible. This proves the puzzle is unsolvable - bacteria cannot clear the 16-point box, the 9-point box (weights sum to 3.0625), and can barely clear an 8-point configuration only with infinite time.