/** Determines where to insert tile, in its current orientation, into game board
   *  @param tile the tile to be placed on the game board
   *  @return the position of tile where tile is to be inserted:
   *           0 if the board is empty;
   *          -1 if tile does not fit in front, at end, or between any existing tiles;
   *          otherwise, 0 = position returned = board.size()
   */
  private int getIndexForFit(NumberTile tile)
  {
    if (board.size() == 0 || tile.getRight() == board.get(0).getLeft())
      return 0;

    if (tile.getLeft() == board.get(board.size() - 1).getRight())
      return board.size();

    for (int i = 1; i < board.size(); i++)
    {
      if (tile.getLeft() == board.get(i-1).getRight() &&
                      tile.getRight() == board.get(i).getLeft())
        return i;
    }

    return -1;
  }

  /** Places tile on the game board if it fits (checking all possible tile orientations if necessary).
   *  If there are no tiles on the game board, the tile is placed at position 0.
   *  The tile should be placed at most 1 time.
   *  Precondition: board is not null
   *  @param tile the tile to be placed on the game board
   *  @return true if tile is placed successfully; false otherwise
   *  Postcondition: the orientations of the other tiles on the board are not changed
   *  Postcondition: the order of the other tiles on the board relative to each other is not changed
   */
  public boolean insertTile(NumberTile tile)
  {
    for (int count = 1; count <= 4; count++)
    {
      int i = getIndexForFit(tile);
      if (i >= 0)
      {
        board.add(i, tile); 1
        return true;
      }
      tile.rotate(); 2
    }
    return false;
  }

1. Inserts the tile properly into the list -- no need to write your own code for that.
2. The fourth call to rotate is just to simplify the code.