Maximizing Neighboring Count in a Sequential Grid Placement: A Combinatorial Optimization Problem

Question

Given a 20x20 grid and an initial value of N = 0, we follow a certain strategy to place a piece into each cell of the grid sequentially. Upon each placement of a piece, we calculate the number of pieces within the eight neighboring cells (top, bottom, left, right, and the four diagonal directions) of the newly placed piece, and add this number to N. What placement strategy should be employed so that when the grid is fully populated, the value of N is maximized?

Answer 1

To maximize the value of N, the placement strategy should prioritize placing the pieces in such a way that each new piece maximizes the number of neighbors it has. In other words, we want to place the piece in a cell that is surrounded by the fewest number of already placed pieces. This approach aims to maximize the contributions of each new piece to N.

To achieve this strategy, follow the steps below:

1. Start by placing the first piece anywhere in the grid (e.g., the center cell).

2. For each subsequent piece to be placed, calculate the number of neighbors (pieces already placed) around each empty cell (unoccupied cell) in the grid. Choose the empty cell with the fewest number of neighbors among its eight adjacent cells.

3. Place the new piece in the chosen empty cell and update the number of neighbors for each surrounding cell (increment their neighbor count by one).

4. Repeat steps 2 and 3 until the grid is fully populated.

By employing this strategy, you ensure that each new piece maximizes its contribution to N by minimizing the number of neighbors it has during placement. As a result, the final value of N will be maximized when the grid is fully populated.