What is the name of old simplified chess-like board game that had pieces shaped like knights and kings such each square arrows indicating where your piece could move?

The game you're describing is likely Shatranj, the predecessor to modern chess.

While the exact design of the board and pieces varied across different regions and time periods, some key features of Shatranj align with your description:

* Knights and Kings: Shatranj featured pieces like the King (often called the "Shah" in Persian), the Vizier (equivalent to the Queen in modern chess), the Elephant (Bishop), the Knight (often depicted as a horse or a rider), the Rook, and the Pawn.

* Arrow-like Movements: Some versions of Shatranj boards may have had symbols or arrows indicating the movement possibilities for each piece. This was more common in instructional diagrams than on actual game boards.

* Simplified Rules: Compared to modern chess, Shatranj had simpler rules and less freedom of movement for some pieces. For example, the Queen was not as powerful, and the Knight's movement wasn't as flexible.

Important Note: It's challenging to pinpoint a single "old simplified chess-like board game" with arrows on the board, as the game evolved significantly across cultures and time. Shatranj, however, fits the general description you've provided.

If you have more specific details about the game, such as its origin, the material used for the board and pieces, or specific names of the pieces, it could be easier to identify the game more precisely.