an 8 by 8 chessboard initially empty, we have 64 identical pieces, and start placing them one by one with the restriction that we can only place a piece on a square if: -the square is empty, and -the square is next to at least three empty squares. what is the largest number of pieces that we can place onto the chessboard? ▍




To solve this problem, we need to maximize the number of pieces placed on an 8x8 chessboard under the given restriction. The restriction is that each piece must be placed on an empty square that is adjacent (orthogonally) to at least three empty squares.

### Step-by-step Analysis

1. **Understanding the Placement Rule**:
   - A piece can only be placed on a square if it is surrounded by at least three empty squares.
   - "Next to" or "adjacent" refers to the squares that share an edge (not a corner) with the square where a piece is placed.

2. **Analyzing Edge and Corner Squares**:
   - Corner squares have at most two adjacent squares (less than three), so placing a piece in any corner is impossible under this rule.
   - Edge squares (not corners) have three adjacent squares at most. After placing a piece here, it can only have a maximum of two adjacent empty squares. Thus, placing a piece on any edge square is also impossible.

3. **Analyzing Interior Squares**:
   - For any square in the interior (not on the edge or corner), it initially has four adjacent squares. A piece can be placed if at least three of these are empty.
   - If a piece is placed on any interior square, it removes two adjacent squares from consideration for future placement (since they will now have fewer than three empty neighbors).

4. **Filling Strategy**:
   - The goal is to maximize the number of pieces placed by ensuring each placement adheres to the rule and leaves the maximum potential for subsequent placements.
   - A checkerboard pattern with every alternate square filled works well because each filled square has at least three empty neighbors.

### Finding the Maximum Number of Pieces

The optimal pattern for filling under the restriction is a **checkerboard pattern**, placing pieces only on squares of the same color (black or white, if colored like a chessboard).

1. **Checkerboard Pattern**:
   - There are 32 black and 32 white squares in an 8x8 chessboard.
   - By filling all squares of one color (either black or white), each placed piece will always have three adjacent empty squares (two horizontally and two vertically).

2. **Counting the Pieces**:
   - If all black squares are filled (or all white squares), this fills 32 squares.
   - This is the maximum number possible without violating the rule, as any additional piece on the opposite color would reduce the number of empty adjacent squares for another square.

### Conclusion

Thus, the largest number of pieces that can be placed onto the chessboard, following the given rules, is **32**.

Would you like more details or have any questions about this explanation?

### Related Questions

1. What would change if the board were a different size, such as 10x10 or 5x5?
2. How does the placement rule affect different chessboard configurations?
3. Can we find a general formula for the maximum number of pieces on any \(n \times n\) board?
4. What if diagonal adjacency counted as well? How would that affect the strategy?
5. How does this problem relate to graph theory and optimization problems?

**Tip:** When solving such constraint-based problems, visualize and consider different patterns and rules to identify optimal solutions efficiently.

An 8 by 8 chessboard initially empty, we have 64 identical pieces, and start placing them one by one with the restriction that we can only place a piece on a square if: the square is empty, and the square is next to at least three empty squares. What is the largest number of pieces that we can place onto the chessboard?

Explore how to maximize the number of pieces placed on an 8x8 chessboard under specific placement rules. This problem involves combinatorics and optimization strategies, perfect for students in Grades 10-12. Discover the solution through a checkerboard pattern strategy.

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