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Ideal Values and Practical Values (part 6):

A Case Study in Piece Values

By Ralph Betza

Basis: Furlrurlbakking

Two of my armies in Chess for Different Armies use the Furlrurlbakking, frlRrlbK, a piece that moves and captures forwards and sideways as a Rook and rearwards (including diagonally to the rear) as a King. (The notation also says it moves sideways as King, but that's moot since the sideways Rook move includes the sideways King move; the extra "rl" in the notation was added merely to make the name of the piece more of an unpronounceable rurlmble.)

According to the simplified counting theory presented in part 1 of this series, the three single-step moves should ideally be fully equivalent to one direction of a Rook's move. I replaced the retreat with short moves not because the short moves were weaker but because retreats are less important over the course of the whole game, and so any difference in the values would be minimized by the lesser importance of retreating moves.

Remember that the Rook gets one third of its average mobility from its ability to move one single square -- in fact, this is why the Furlrurlbakking was capable of being invented even before I had devised the theories of ideal values and simple addition.

The short moves are powerful. The Man, or Commoner is notably stronger in practice than a minor piece. Understanding the power of the WF is important to understanding the strength of the frlRbK.

However, the Rook is a piece whose role in the game is usually to remain on the board until the endgame, when with fewer pieces on the board it is more likely to be able to make the long moves for which it is famous. Likewise, though we start the game with foe in front (which is the reason that forward movement is more important over the whole course of a game), by the endgame the armies are scattered around the board and retreating moves become more common, and forwardness becomes less important.

Ever since Hans Bodlaender pointed this out to me, I've worried about whether the Furlrurlbakking might have an Achilles heel in that combination of 3 things: first, it is a piece of a rank likely to survive to the endgame, second, long movement powers are stronger in the endgame, and third, retreats are less unimportant in the endgame. If this is so, then unless we assume that the frlRbK's choice of three retreats give it enough opening and midgame advantages to compensate for its endgame disadvantage, and there seems to be no reason for this assumption, then it follows that the Furlrurlbakking is weaker than the Rook, possibly enough weaker than the Rook that the difference will often decide the game; and that therefore the armies that use it are fatally flawed. Yes, I worry about this stuff.

The position that shows the frlRbK at its worst has White K on b8 White frlRbK on a7, Black Pa2 and Kb1. A Rook at a7 would draw easily. Instead, the Furlrurlbakking must helplessly watch the Pawn promote.

In October 2001, I finally realized that the Furlrurlbakking at a7 draws easily. It allows the Pawn to promote, and then laughs at the inability of the enemy King to approach it from the rear. The Pawnless ending Q versus R is (in most cases) an easy win, but Q versus frlRbK is merely a draw! I believe the same is true if the opponent has Chancellor rather than Queen.

The Furlrurlbakking is different from the Rook, and it has different strengths and different weaknesses, but I now believe more than ever it has the same strength as the Rook.

The player with the Rook will strive for positions where the Rook shows to advantage, while the player with the Furlrurlbakking will play in a way that emphasizes the frlRbK's strong points, and that's what Chess is.

Subvariations on the Furlrurlbakking

Changing the retreat of the frlRbK to give it some longer jumps would take away its powerful short moves, and the resulting piece would probably lose the Pawnless endgame versus Queen.

Here is an interesting idea which came to me after most of this article was written: How about a piece which moves as R, but captures as Furlrurlbakking? Or captures as R, but moves as the other? And the same for every single one of the Rooklike pieces discussed here? You can see that there is neither space nor time to discuss all the possibilities! However, the general principle is that the value of the piece should be the average value of R and the other piece, and since most of the pieces already have R value or something close, many of the pieces that seem too weak or too strong would be helped by this rule.

First Variation -- the Bisserfoof

The bsRfWfF is the opposite of the Furlrurlbakking -- it moves to the side and rear as Rook, but advances as a King.

The Bisserfoof is not worth as much as a Rook over the course of a whole game, and the reason is most elegantly explained by an example that occurs later in this text. However, what an interesting piece it would be to use!

How much less than a Rook is it worth? Would a forward lame Dabbabah non-capturing move added to its power make it equal to R, or stronger? I wish I knew, and I'd welcome any ideas.

Intuition says that the Bisserfoof is worth nearly as much a a Rook. If only it were possible to know how nearly!

Second Variation -- Fibberking and Sirking

The fbRK moves as King, or advances or retreats as Rook; it does not have the Rook's sideways move. Where the Furlrurlbakking substitutes three short moves for one direction of a Rook's move, the Fibberking substitutes six for two. Given the importance of distance in forwardness, one should feel intuitively comfortable with the idea that the Fibberking is as strong as a Rook, despite its sideways slowness that may sometimes cause it to be caught off base.

However, the six short moves substituted for the sideways Rook moves include two advancing moves! Although they are short, they are forwards, and I wouldn't be at all surprised if the Fibberking were noticeably stronger than the Rook for this reason; but on the other hand, perhaps they merely compensate for the fbRK's sideways imbalance.

Even if the Fibberking is not stronger than a Rook, of course there are situations in which it is stronger and others in which it is weaker. The characteristic attack of the Fibberking is to zoom down an empty file, most likely with a non-capturing move, and arrive at a position where it attacks several pieces at once using its multiplicity of short-range moves. In other words, to attack like a Man, merely arriving by chariot.

The sRK would move as King, or sideways as a Rook. The importance of distance in forward moves should make the practical value of the Sirking less than a Rook although its ideal value is the same as a Rook, or indeed the same as a Fibberking or as a Furlrurlbakking.

According to the theory that balance allows practical values to come closer to ideal values, adding a Knight's power to Sirking or Fibberking, or even to Furlrurlbakking or Rook, would produce new pieces whose ideal values would all be Queen and whose practical values would be closer to each other than the original pieces.

In other words, the difference in value between Sirking+Knight and Fibberking+Knight should be less than the difference in practical value between sRK and fbRK because the added N move gives each one more movement in the direction it lacked. Balance.

Of course, while reading this you have already imagined a piece that alternates between Fibberking and Sirking, that is, a piece which turns ninety degrees each time it moves. Now you know its value. Feel free to name it and use it.

Third Variation -- Nine Short Moves plus Unidirectional Rook

Firking and Birking seem much more natural than Lurking and Rurking, don't they? ("If I ever saw anybody lurking and rurking, I'd call a cop!" rim shot on the drum, the audience laughs.)

There is a certain symmetry, though, which gives an insight into why the Furlrurlbakking is worth more in practice than the Bisserfoof. Imagine that you start the game with rRK on h1 and lRK on a1, while your opponent started with rRK[2] on h8 and lRK on a8 -- wouldn't you feel that you had a big disadvantage? Starting with the Bisserfoof is in principle very much the same as starting with rRK on h1, and that's the insight. So obvious now that we know it.

The brK's ability to make long moves cannot be used until it has climbed to the other side of the board, while the frK's long moves are on the board from the start of the game (though they may be blocked by Pawns and other obstructions). The brK is worth less (but again, it would be an interesting piece to use.)

The fRK, bRK, lRK, and rRK are only seven short moves plus one direction of a Rook's move, and so each is ideally worth less than a R by one sixth of a Rook, enough to decide the game.

WDH is a three square x-ray Rook, and because of the way these powers work together I think the piece is much too strong to use as a Rook. Even as a sideways move, it seems too strong; but as a retreat, perhaps it would be okay.

Except for the special combination of WDH in the same direction, and with the additional exception that W, D, and H movements in the same direction as Rook's movement can't be chosen, one could start with a unidirectional Rook, add any nine "short" moves from N, W, F, A, H, or D, or even Bishop, where one direction counts as two short moves, in other words any nine subatomic "move and capture" directions from the standard list of atoms that are useful on an 8x8 board (of course you could also have some movement powers different from capture powers), and if the combination of ingredients you pick is well-balanced the resulting piece should be a Rook in practical value as well as in ideal value.

That's thousands of possible new Rook-valued pieces! (Some mathematician will respond that there are really only 932 possible combinations, or some other number. It's too much work for me to figure out right now, I'd have to derive or remember how.)

For example, fRbWN, a very nice piece. Too nice, perhaps, because adding the forwardness of the N to the forward Rook presumably makes the Firboon too strong, a rare case where practical value is greater than ideal value.

bBfRbWsfNbbN is a much better balance. sfNbbN is the Barc, and so the Bifferbubarc (accent on biff and boo so that the name can be used in Shakespearean sonnets or in Rubaiyat[1]) is a forward Rook plus retreating Bishop plus wide forward Knight plus narrow rearward Knight. It has a Rookish ideal value, and I think it is well-balanced enough that it is a Rook in practical value.

Twelve Short Moves

I have no fear of fighting against a Rook with an NW, NF, ND, or NA, (or NH), because the Knight move is usually "long enough" on an 8x8 board (plus, my piece moves in so many directions!), but with WFD or WAF, I feel a slight degree of discomfort -- and this despite the fact that Wooft and Waff "can mate", but some of the Augmented Knights cannot.

However, this is just chessplayer's intuition, the same intuition that worried bootlessly about the Furlrurlbakking's possible endgame weakness. Perhaps WFD and WAF are okay -- but WAF seems so awkward in the opening, I've never liked it much.

The WFD seems to be too concentrated on very short moves. The ffNsbNWF -- the Crabman! -- is likely to be better balanced because it has some "long enough" moves.

The WAD seems a bit dispersed, but I believe in its value despite the following endgame study: W has Ka1 and WAD on e5, Black has Kd4, White's move. If the White K can protect the WAD, Black will be checkmated. With a WD instead of a WAD, the game is drawn. White wins, but best defense makes it surprisingly difficult. (If you find the one and only correct way to win on the first try, it might not seem so hard -- try to find a second way!)

The FAD is colorbound and has been tested and is known to be somewhat inferior to a Rook -- but not by very much. Note that the FAD jumps to all same-color squares in its range, while the NW jumps to all opposite-color squares in the same range. Too bad there isn't room in one army for two pairs of Rook-valued pieces unless do something funny with the value of some other piece. The Colorbound Clobberers did this successfully, but it's dangerous to have a strange material balance.

HFD is the beloved Half-Duck, a lovely piece and well worth a Rook, while WFD is mentioned above in this article.

Many other combinations of the standard ingredients exist, but have not been tested, have not been named, have not been used in games. Think of this as an opportunity!

Summing Up

If the Forlrurlbakking has a weakness, it is that all its moves in one direction are short -- in other words, a question of balance.

I don't think the frlRrlbK has a significant weakness. I think it is adequately balanced because it has long moves in some directions, short in others.

Applying the principle that created the Forlrurlbakking, replacing one direction of the Rook's move with three short moves, we can derive new pieces with two or one Rook directions; and when we get down to zero Rook directions, we're talking about some well-known and well-tested pieces in addition to many unknown and untested possible pieces.

This exercise seems to support the idea that we can simply add pieces and parts of pieces, and when the total reaches a certain number we get a Rook-valued piece.

But that's a piece whose ideal value is Rookish. For many of the pieces I derived, I was able to argue that the piece was in practice weaker or stronger than R. I hope my arguments convinced.

One principle allows us to simply stitch together a bunch of piece parts -- Igor, we need the brain of a Rook and the heart of a Knight -- and create something new of predictable value.

That principle is -- audience let me hear you, all together now -- b l c . I can't hear you! Let's try again, just the left half of the auditorium, it's bal . That was weak, I'll bet you folks on the right can do better, give it to me now, it's ance! Okay all at once, let's hear it for


[1] "The Bifferbubarc moves, and having moved," for example.

[2] The opponent's right is your left...

Click here for the previous article in this series.

Written by Ralph Betza.
WWW page created: November 3rd, 2001.