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A princess (also known as a cardinal, archbishop, 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.
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The princess 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.
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 checkmate a lone king all by itself if the opposing player blunders by putting his king on a corner square where the princess can checkmate it two squares away diagonally, while the rook+knight compound cannot checkmate a lone king by itself.
The amazon was first used in Turkish Great Chess, a large medieval variant of chess, where it was called the vizir (not to be confused with the piece more commonly referred to as the wazir today, which is the (1,0) leaper). 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.
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Ralph Betza (inventor of chess with different armies, in which the princess was used in one of the armies) rated the princess 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 versus 8 directions for queen.
Computer self-play studies show that a single pawn is enough to compensate the difference between queen and princess on an 8x8 board, and that on 10x8 boards princess plus pawn even have a light advantage over queen. This is somewhat surprising, as the value difference of the components (rook vs knight) is closer to two pawns, implying an unusually large synergy between the bishop and knight move. Although princess versus rook is a usually a draw, so is queen versus princess. King and princess versus king is a forced win for the side with the princess; checkmate can be forced within 17 moves. In comparison, the queen requires 10 moves and the rook requires 16.