In the first case there is a board with dimensions 2*2 with 4 squares inside it of 1*1 dimension. The first square is empty i.e. it is a missing square in the upper left corner, so the L-shaped arrangement for this case will be LR and it will be filled at the remaining coordinates as it has the corner for the L-shaped arrangement at the lower right. Similarly, In the second case a missing square is there in the upper right corner so the L-shaped arrangement for this case will be LL and it will be filled at the remaining coordinates as it has the corner for the L-shaped arrangement at the lower left, in the third case a missing square is there in the lower right corner so the L-shaped arrangement for this case will be UL and it will be filled at the remaining coordinates as it has the corner for the L-shaped arrangement at the upper left and in the fourth case a missing square is there in the lower left corner so the L-shaped arrangement for this case will be UR and it will be filled at the remaining coordinates as it has the corner for the L-shaped arrangement at the upper right

We will use x coordinate and y coordinate to record the location of the missing square. It is convenient to treat the origin (0, 0) as the lower left corner of the board.  import java.util.Scanner;

public class TrominoTiling {
    private static String[][] board;

    public static void main(String[] args) {
        Scanner scanner = new Scanner(System.in);
        System.out.print("Please enter size of board (need to be 2^n and 0 to quit): ");

        int boardSize = scanner.nextInt();
        if (boardSize == 0) {
            scanner.close();
            return;
        }

        System.out.print("Please enter coordinates of missing square (separated by a space): ");
        int xMissing = scanner.nextInt();
        int yMissing = scanner.nextInt();

        // Initialize the board
        initializeBoard(boardSize);

        // Mark the initially missing square
        board[yMissing][xMissing] = "MS";

        // Fill the board with trominoes
        fillBoard(0, 0, boardSize, xMissing, yMissing);

        // Print the result
        for (int y = boardSize - 1; y >= 0; y--) { // Start from the top for display
            for (int x = 0; x < boardSize; x++) {
                System.out.print(board[y][x] + "\t");
            }
            System.out.println();
        }

        scanner.close();
    }

    private static void initializeBoard(int boardSize) {
        board = new String[boardSize][boardSize];
        for (int i = 0; i < boardSize; i++) {
            for (int j = 0; j < boardSize; j++) {
                board[i][j] = "  ";  // Initialize with spaces
            }
        }
    }

    private static void fillBoard(int x, int y, int size, int xMissing, int yMissing) {
        if (size == 2) {
            // Place the trominoes around the missing square.
            String trominoMarker = "UR"; // Default placement, adjust based on actual position of the missing tile.
            for (int i = 0; i < size; i++) {
                for (int j = 0; j < size; j++) {
                    if (x + i != xMissing || y + j != yMissing) {
                        if (xMissing == x && yMissing == y) trominoMarker = "LR"; // Lower Left
                        else if (xMissing == x + 1 && yMissing == y) trominoMarker = "LL"; // Lower Right
                        else if (xMissing == x && yMissing == y + 1) trominoMarker = "UR"; // Upper Left
                        else if (xMissing == x + 1 && yMissing == y + 1) trominoMarker = "UL"; // Upper Right
    
                        board[y + j][x + i] = trominoMarker;
                    }
                }
            }
            return;
        }
    
        int halfSize = size / 2;
        // Identify center for placing the tromino if the missing square is not in this quadrant.
        int centerX = x + halfSize - 1;
        int centerY = y + halfSize - 1;
    
        // Determine in which quadrant the missing square resides.
        boolean missingInLowerLeft = xMissing < x + halfSize && yMissing < y + halfSize;
        boolean missingInLowerRight = xMissing >= x + halfSize && yMissing < y + halfSize;
        boolean missingInUpperLeft = xMissing < x + halfSize && yMissing >= y + halfSize;
        boolean missingInUpperRight = xMissing >= x + halfSize && yMissing >= y + halfSize;
    
        // Place the center tromino based on the quadrant of the missing square.
        if (!missingInLowerLeft) board[centerY][centerX] = "MS"; // Bottom-Left
        if (!missingInLowerRight) board[centerY][centerX + 1] = "MS"; // Bottom-Right
        if (!missingInUpperLeft) board[centerY + 1][centerX] = "MS"; // Top-Left
        if (!missingInUpperRight) board[centerY + 1][centerX + 1] = "MS"; // Top-Right
    
        // Recursively solve for each quadrant.
        fillBoard(x, y, halfSize, missingInLowerLeft ? xMissing : centerX, missingInLowerLeft ? yMissing : centerY);
        fillBoard(x + halfSize, y, halfSize, missingInLowerRight ? xMissing : centerX + 1, missingInLowerRight ? yMissing : centerY);
        fillBoard(x, y + halfSize, halfSize, missingInUpperLeft ? xMissing : centerX, missingInUpperLeft ? yMissing : centerY + 1);
        fillBoard(x + halfSize, y + halfSize, halfSize, missingInUpperRight ? xMissing : centerX + 1, missingInUpperRight ? yMissing : centerY + 1);
    
        // Replace the "MS" used for simulations with correct tromino markers.
        if (!missingInLowerLeft) board[centerY][centerX] = "LR"; // Center tromino facing Lower Right
        if (!missingInLowerRight) board[centerY][centerX + 1] = "LL"; // Center tromino facing Lower Left
        if (!missingInUpperLeft) board[centerY + 1][centerX] = "UR"; // Center tromino facing Upper Right
        if (!missingInUpperRight) board[centerY + 1][centerX + 1] = "UL"; // Center tromino facing Upper Left
    }
    
    
} Please enter size of board (need to be 2^n and 0 to quit): 4
Please enter coordinates of missing square (separated by a space): 1 3
UL      MS      LR      LR
UL      UL      UL      LR
UL      LR      LL      UR
UL      UL      UR      UR this is the output that i got and the expected output is Please enter size of board (need to be 2^n and 0 to quit): 4
Please enter coordinates of missing square (seperated by a space): 1 3
LL MS UR UR 
LL LL LR UR 
LL LR LR LR 
LL LL LR LR