Fairy ChessY'know, the game designer in me always kinda wanted to invent new chess peices, and got sad when nobody wanted to play with those pieces. Turns out an awful lot of pieces have been invented (and many played in historical versions of chess).
There's actually a fuckton of these, but I'll go over the ones famous enough to have wikipedia pages.
The AmazonCan move the same as a queen or a knight. Piece is super busted.
Estimated material value...11.5 to 14.
Empress (or Chancellor)This one can move like a rook or a knight.
Estimated material value...8-9. (Usually 0.5 to 1 below a queen).
Some interesting commentary, though, is that this piece's value goes up a lot as the board opens up. In endgame situations, some people like to promote to a Empress/Chancellor over a queen.
Princess (or Archbishop)Can move like a knight or a bishop. Technically capable of checkmating a king with the help of no other pieces if the king moves to the corner square. (Although for the most part, better early in the game rather than in endgames, doesn't really have a way to force the king into the corner)
Estimated material value...7.7-8.75
I'm actually super interested in the material value here, cause it's typically seen as almost as good as an Empress/Chancellor, and not even that far behind a Queen. And considerably more than a Knight plus a Bishop. The synergy between the diagonals and the knight moves is pretty big here. Covering a ton of the 2x2 square around the piece, and enabling it to move between light squares and dark squares (one of the key things that makes bishops worse than rooks--only being able to reach half the squares on the board).
The other interesting thing is that the piece valuation of a weakened princess (The Leopard--same piece, but the bishop movement has a maximum movement of 2) was determined by computer simulation to be barely worse than a princess (computer simulation had 7.7 pawns for the princess, and 6.7 pawns for the leopard).
The Alfil (or elephant)Old historic piece that jumps two spaces diagonally and nothing else.
Estimated Value...1.5
Yeah, this piece is pretty bad, unable to ever reach 75% of the squares on the board.
Camel (or Long Knight)Historic piece that jumped 3 up 1 left.
Estimated Value...2
Similar to a Knight except...it's a bit like a bishop in that it can't leave the dark squares and...that brings the value down.
DabbabaAnother historic piece. Jumped 2 spaces horizontally or vertically.
Estimated value...1.5
Just like the diagonal jumper...75% of the board is literally unreachable for this piece, worth about the same value.
FerzA historic piece that was later replaced by the queen (what an upgrade lol). Could move one space diagonally only.
Estimated value...1.5
Moves really slowly and awkwardly, but at least it can reach...half the board.
MannLiterally just moves like a king without actually being a king (can't castle, losing it does not lose the game).
Estimated value...3
Thought to be worth a little less than a Knight early in the game, because developing it is slow and awkward, but worth a little more than a Knight in endgames.
Nightrider (also sometimes called knightmare or unicorn)So...this is a knight that can just keep moving in a straight line, as long as that straight line is made up of the same Knight move.
Estimated value...5
I find it very hard to look at a chess board and see how to move this piece to set up checkmates. Like...taking a step that's 2 knight moves away, in order to block a square that's 3 knight moves away from the square you just moved to...I can count out the spaces, and confirm that the moves work, but...wow, my brain does not visualize that. Maybe it would get easier with practice.
Wazir (or Vazir)Can move one space horizontally or vertically
Estimated value...1-1.5
Despite being by far the slowest piece (slower than the Ferz) at least it can reach every square. Worse than these other low value units early on since it develops so slow, but goes up in value as the game progresses.
ZebraHistoric piece going back as far as the 13th century. Jumps one space horizontally and two spaces diagonally.
Estimated value...1.75
Despite the fact that it doesn't have the problem of the Camel where the Camel is restricted to one colour, nonetheless considered worse than a Camel, cause it actually just moves too far for an 8x8 board. Some theorize it'd be worth about the same as a Knight on a 10x10 board.
As a component of other pieces (like the amalgam pieces such as princess) it's considered to add about as much value to the piece as a knight.
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Modern versions of fairy pieces mostly seem to aim for pretty good pieces (like...typically somewhere between a rook and a queen in value).
A couple examples from one variant (musketeer chess)
https://musketeerchess.net/p/games/musketeer/rules/rules-short.phpOther than just some historic strong pieces (Princess, Empress, Amazon...which they chose to use the name of Archbishop, Chancellor, and Dragon)
A couple examples of pieces they have...
"unicorn" (Knight+Camel, so 1 to the side, and 2 or 3 in the other direction).
"elephant" (Jumps 1 or 2 horizontally, vertically, or diagonally)
"hawk" (Jumps 2 or 3 horizontally, vertically, or diagonally)
"Leopard" (toned down princess. 2 move bishop + knight)
"Spider" (2 move bishop + knight + Dabbaba a.k.a. can also jump 2 horizontally/vertically).
Plus a few other pieces that I like the design of less as they move asymmetrically.
Values of these new pieces (according to their own website) range from 5.5-8.15. These...do appeal to me (in the way a lot of older pieces do not--not hugely excited about the old historical pieces that are worse than a king). Although...I can't help but feel the number of pieces they're adding is a bit inflated just to sell product.
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Anyway, Amazon, by most reports, is so powerful it's not actually fun to play with.
...But a few of these other medium-high pieces I would be interested in experimenting with. The historical Princess and Empress look pretty neat. The musketeer pieces that aren't historical...vary--I feel like they have more pieces than they need just to sell more product. But at very least their "unicorn" and "hawk" designs are pretty neat. Does capture the idea of an upgraded knight (while also each being different from each other). And sure...I could see the logic behind mixing in one of the powerful 2 move pieces (between "Leopard"/"Elephant"/"Spider").
Actually, I'm kind of surprised they don't have a piece that's just "jump to any square that's 2 spaces away". (In terms of combining old pieces, that would be Knight+Dabbaba+Alfil). Just seems like an easier to explain version of a 2 moving power piece. Seems like a bunch of these pieces control 16 squares, so that should fit right in the power band.