Two players are playing a turn based game on a $n \times n$ chessboard. At the beginning, only the bottom left corner of the chessboard contains a piece. At each turn, the player moves the piece to either the square just above, or the square just right, or the diagonal square just right-top. If a player cannot make a move, he loses the game. The game is played once on each $6\times 7$, $6 \times 8$, $7 \times 7$, $7 \times 8$, and $8 \times 8$ chessboard. In how many of them, can the first player guarantee to win?

$ \textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ \text{None}
$
1. **Understanding the Game Mechanics**:
   - The game is played on an \( n \times n \) chessboard.
   - The piece starts at the bottom-left corner \((1,1)\).
   - Players take turns moving the piece to either the square just above, the square just right, or the diagonal square just right-top.
   - The player who cannot make a move loses the game.

2. **Identifying Winning and Losing Positions**:
   - The game ends when the piece reaches the top-right corner \((n, n)\).
   - If a player moves the piece to \((n, n)\), the other player wins because the first player cannot make a move.
   - We need to identify positions from which the first player can guarantee a win.

3. **Analyzing the Board**:
   - The first player loses if they are forced to move to a position from which the second player can always win.
   - We need to identify these losing positions.

4. **Pattern Recognition**:
   - The losing positions are those from which any move leads to a winning position for the opponent.
   - For a \( n \times n \) board, the losing positions are those where both dimensions are odd. This is because the first player will always be forced into a position where the second player can mirror their moves and eventually force them into a losing position.

5. **Applying the Pattern to Given Boards**:
   - \(6 \times 7\): One dimension is even, so the first player can win.
   - \(6 \times 8\): Both dimensions are even, so the first player can win.
   - \(7 \times 7\): Both dimensions are odd, so the first player loses.
   - \(7 \times 8\): One dimension is even, so the first player can win.
   - \(8 \times 8\): Both dimensions are even, so the first player can win.

6. **Conclusion**:
   - The first player can guarantee a win on the \(6 \times 7\), \(6 \times 8\), \(7 \times 8\), and \(8 \times 8\) boards.
   - The first player cannot guarantee a win on the \(7 \times 7\) board.

The final answer is \(\boxed{4}\)