It moves forward as a narrow Knight, rearward as a wide Knight, can reach every square of the board, and is obviously worth half a Knight. Perhaps it's worth a tiny tiny bit more or less than half, but we can't be that precise.
Here is the ugly ASCII diagram of its move:
. . . . . . . . . . . . . . . . . . . 1 . 1 . . . . . . . . . . . . . . X . . . . . 1 . . . 1 . . . . . . . . . . . . . . . . .
Does it look like a crab to you? It does to me; I see its pincers held out front while scuttles sideways on its little legs.
The diagram of how many moves it takes to get to each square reveals a very strange pattern:
8 5 2 5 2 5 2 5 3 6 3 6 3 6 3 6 4 7 4 1 4 1 4 7 5 2 5 2 5 2 5 2 6 3 6 3 X 3 6 3 7 4 1 4 7 4 1 4 2 5 8 5 2 5 8 5 9 6 3 6 9 6 3 6
The Crab is very rare because it is a half-Knight that is likely to be worth half a Knight as a piece all by itself. Of all the natural half-Knights (F, W, D, A, narrow Knight, wide Knight) only the W can see the whole board, and the W is terribly short-range and awkward, and only the F is probably worth as much as a half-Knight (despite the fact that its range is short and that it is colorbound, the Ferz is flexible, and stronger than you might think).
You can construct some other half-Knights, ffNbF or ffNbsL for example, but they seem to have some practical disadvantages; and so, the Crab is the "best" half-Knight.
Of course, the fsNbbN -- the backwards Crab -- is a bit weaker than the Crab, and a bit clumsier as well.
The best thing about the Crab is that it is similar in value to the Ferz but has a much more interesting pattern of movement. In other places, we have found the Ferz to be a useful piece when we want to have a weak piece as a carrier for powerful, strange, and exotic powers, but the Ferz is kind of boring -- using the Crab instead should be a lot better.
Let's look at each of these things in greater detail.
The RffNbsN (Rook plus Crab) is an interesting new piece whose value is midway between Rook and Queen (and should be the same as the value of a RF or RA), and the ffNNbsNN (Crab-Rider, makes a Crab move and then if that square was empty it has the option of continuing in the same direction; for example, ffNNbsNNd1-g7 is legal if both e3 and f5 are empty) is a piece of completely unknown value.
Thanks to the Crab (and the Lobster, bFfA), it is also more possible now to design some chess variants with weaker pieces. I don't remember exactly what, but I do remember that there were a few things I gave up on because there weren't so many decent pieces that were weak enough....
A Queen is worth more than a Rook and a Knight; does this mean that a Knight is worth more than two Crabs? (If the answer is yes, the difference is probably less than a third of a Pawn, and can be ignored in most positions.)
If one player has two Crabs instead of Knights, and an Amazon Queen instead of the regular Queen, this is probably an improved version of the Amazon Army.
Here are two puzzles; the first is easy:
Two Crabs cannot defend each other. How many Crabs does it take to form a mutually-defended chain?
In the endgame with King and Crabs versus bare King, how many Crabs are needed to force checkmate?
Suppose that your Knights were replaced by Crabs with relay powers (any friendly piece defended by a Crab can move (or capture) like a Crab; once it moves away from contact with the Crab, it loses this extra power; in some games Pawns can gain power this way). Using the rule that Pawns may not get relay powers, I think that you have a disadvantage, but a very small one.
So you see, the Crab is a good piece to act as the carrier of powerful powers.
The Lobster (bFfA) is another piece from my old notes: it moves forward as Alfil, to the rear as Ferz. Perhaps it is even weaker than Ferz, but its extra distance makes it seem like a much nicer piece to have. This feels like a natural piece.
A bFfD moves in only 3 directions, but is interesting.
bsWffN is probably better (stronger, and feels more natural) than bFffN; bsWfA is a stronger Lobster that's not colorbound (but it's too similar to the Lobster). bsWfD is a candidate to replace bFfD, but bFfD's move has such an interesting shape....
Unfortunately, bFfD is probably only worth a Pawn, (The forward-only move is half the value of the D, the rearward F-move is worth a quarter or a third of an F, and the piece is colorbound) and if it starts on the first rank it must go froward or be stalemated. I like the bFfD, but for the present purpose, I'll have to use the WfD instead: a bit stronger than a Crab or a Lobster, the Snail (WfD) is going to be our Rook. Or maybe the WfR3, non-jumping short Rook when it moves forwards?
For a Queen, we can add the Snail to either the Lobster or the Crab. I like the Crab a lot better for this, so the Queen is WfDffNbsN, whose funny notation looks awful but whose move isn't so bad, nor hard to remember -- it's the CrabSnail, worth more than a Knight but less than a Rook.
Should Pawns be allowed to promote to WD, which leaps one or two squares Rookwise in all directions, in addition to the other pieces? (It is much easier to checkmate a bare King with a WD than with a CrabSnail, even though the CrabSnail is generally the stronger piece.)
Instead of this rule, I have chosen to allow victory by bare King.
Now we have a whole army of weak pieces, less than half as strong as the FIDE-Chess army. It is possible to play a game of chess where both sides use this army:
A game of DemiChess should take more moves than a game of FIDE Chess, but require approximately the same amount of thinking time; therefore it is better suited to face-to-face or internet-server play than to correspondence play.
Most of your moves will be easy to make, and you will think more about strategy than about tactics, but when you do think tactically, you may find yourself analyzing 10 or 20 moves in advance.
You may notice that there are enough alternate pieces suggested in the original discussion to make up a whole new army, so you could have DemiChess with Different Armies.
The usual bunch of problem tasks are worth looking at: What's the foolsmate? What's the shortest possible game ending in perpetual check? What's the shortest game ending in stalemate? Can you win a Pawnless endgame with a CrabSnail versus two Lobsters?
Other DemiChess variants are sure to be interesting games, but nothing special: Cylindrical DemiChess, Alice DemiChess, Dynamo Demichess, guaranteed good, but only good, not specially good.
Some DemiChess variants just won't work: 8x8 Race Demichess fails because of 1. (ffNbsN)g1-f3+, for example.
For example, giving Relay power to all the pieces would make the army stronger. According to a guess I published elsewhere, it wouldn't be enough; but it was a guess, and nobody has tested it yet. I'm willing to guess that my other guess was wrong, and that a DemiChess Relay army would be roughly as strong as the FIDE army.
Alternatively, I have guessed at the strength of rifle capture (when you capture a piece, your piece doesn't move; instead, it stands still and "shoots" the piece it wants to capture, which simply disappears from the board and leaves an empty square). A DemiChess Rifle Army might well be the correct strength.
A DemiChess army of doublemove pieces would be much too strong; likewise an Iron DemiChess Army. I need to choose one, so here goes:
1. e4 d5 2. e4:d5 Q:d5 3. (WfDffNbsN)d1-e3?! Qd5-b5?! 4. b3! e6 5. (ffNbsN)b1-c3 Qb5-b6 6. (ffNbsN)g1-f3 Ng8-f6 7. (bFfA)f1-d3 Bf8-c5!? 8. (WfDffNbsN)e3-e2 Nb8-c6 9. (bFfA)c1-a3 Bc5-d6 10. (bFfA)d3-b5 O-O 11. (bFfA)b5-c4 Rf8-e8
With 2...Q:d5, black attacked g2; 3. (ffNbsN)g1-f3 is probably a better way to defend it. 4.b3! is the only way to defend b2, because (bFfA)c1-a3 allows a later Bf8:a3. Finally, (bFfA)b5-c4 threatens to shoot e6, but Rf8-e8 would take revenge on the Queen.
This position looks promising for White and makes us think the DemiRifleChess army is too strong, but if Black had played 9...Bc5:a3 10. (WfD)a1:a3, the exposed WfD would have been so weak that we would think the DemiRifleChess army was too weak.
A day later, I think that the list of moves that are probably better than those played in the sample should include 1...Nc6, 2. e5, and 11...Nc6-a5, in addition to those already mentioned.
All in all, I think this army looks good enough to try, and I'd consider it an "official" army for chess variants with different armies. It's possible that you could play several games, learn the best strategies for handling this type of battle, and discover that one side or the other has a winning advantage; however, starting with ignorance, the armies seem evenly matched.
Both plain old DemiChess and DemiRifleChess are guaranteed to be especially good games, and Progressive DemiChess is strongly suggested.