An archbishop (also known as a cardinal, princess, or simply bishop–knight compound) is a fairy chess piece that can move like a bishop or a knight. It cannot jump over other pieces when moving as a bishop, but may do so when moving as a knight. Below, it is given the symbol NB from Betza notation.
History and naming 
The archbishop is one of the most simply described fairy chess pieces and as such has a long history and has gone by many names. A generic name would be the bishop–knight compound. The most commonly used names for this piece are the cardinal, archbishop and princess.[note 1]
The name archbishop was introduced by José Raúl Capablanca in his large variant Capablanca chess. He originally called it the chancellor, but later changed the names and the rook–knight compound became known as the chancellor. Both of these names refer to higher ranks than the bishop in the Roman Catholic Church, but archbishop does so more obviously to most people, and thus became more popular. In fact, the name archbishop has been used for other augmented bishops as well, such as the "reflecting bishop" (which reflects off the sides of the board) and the bishop–king compound. A similar approach was taken by Christian Freeling, the inventor of Grand Chess, who named it the cardinal. Both archbishop and cardinal are popular names for the bishop–knight compound.
The name princess is more widely used among problemists. By analogy with the queen, which is a rook–bishop compound, it was decided that the three basic combinations of the three simple chess pieces (rook, knight, and bishop) should all be named after female royalty. Since the bishop–knight compound seemed to be obviously weaker than the rook–knight compound (as the bishop is weaker than the rook), the name princess was used for the bishop–knight compound and the rook–knight compound was called the empress. However, the bishop–knight compound can get (but not force) checkmate of a lone king by itself (with the king in a corner and the attacking archbishop two squares diagonally away), while the rook–knight compound cannot.
The archbishop was first used in Shatranj al-Kabir, a large variant of chess from around the 14th century, but its name in that game is unknown. It was introduced in the West with Carrera's chess, a chess variant from 1617, where it was called a centaur, and has been used in many chess variants since then.
Ralph Betza (inventor of chess with different armies, in which the archbishop was used in one of the armies) rated the archbishop as about seven points, intermediate between a rook and a queen, noting that it was "a weak Queen" and that its value was increased by its 12 different directions of movement.
The archbishop is closer in strength between the rook and the chancellor (rook–knight compound) in the endgame. Although archbishop versus rook is a usually a draw, so is queen versus archbishop. King and archbishop versus king is a forced win for the side with the archbishop; checkmate can be forced within 17 moves. In comparison, the queen requires 10 moves and the rook requires 16.
See also 
- Less common names the piece has acquired include adjutant, aircraft, centaur, chancellor, davidson, deacon, equerry, fox, horseman, janus, monk, pilot, police chief, prime minister, rhino, squire, superbishop, templar, wazir, and zek.
- Piececlopedia: Bishop–Knight Compound at The Chess Variant Pages
- Endgame statistics with fantasy pieces at The Chess Variant Pages
- The NB (Cardinal) by Ralph Betza
- BuyPoint Chess by Ralph Betza. Contains a list of pieces and approximate values for each of them.