in gothic chess yet??? (yup I want fame :))

To derive a nice formula for the strength of a Rook on a board of Y height and Z width, here is what you do.

First, examine the geometry of the board. You can see that from the 4 corners, the King can be safely checked a total of (Y - 2) + (Z - 2) times. Recall is a Rook is next to the king while delivering check (like Y -1 or Z - 1) then the king can just capture it.

So, we have 4((Y-2)+(Z-2)) so far.

Basically, in the corner, the king can recapture 2 of the rook checks.

Now move over one square for the king (b8) or move down one square (a7). The king would be able to recapture against 3 rook checks. In the case of the a7 king, it could capture a Rook checking on a8,a6, or b7.

You notice on any rectangular board, there are 4 pairs of squares where this is true. On the 8x8 chessboard these are b8/a7, g8/h7, b2/a1, and g2/h1. So, we have 4 instances of (Y-2) + (Z-1) and 4 instances of (Y-1) + (Z-2).

To 4((Y-2)+(Z-2)) from the 1st calculation we add 4((Y-2) + (Z-1)) + 4((Y-1) + (Z-2)).

Recall a probability is a quotient, that being these total squares of safe check divided by the entire population of arrangements. After placing one piece on the board, there are ZY - 1 slots remaining for the next piece. But the first piece can occupy any one of those ZY squares, so you get fractions in terms of ZY-1 and ZY(ZY-1) when you compute the probabilities.

When you collect all such terms for the rook, you get:

P(safe check) = Z + Y - 6/(ZY - 1) + 2(Z + Y)/[ZY(ZY-1)] for the probability that a Rook can safely check a king on any such size board.

Plug in Y = Z = 8, and you get:

8+8-6/63 + 2(8+8)/64(63) =
10/63 + 32/64x63 =
640/64x63 + 32/64x63 =
672/4032 = 1/6

So my results match Taylor, I just computed it a different way since Gothic Chess has a rectangular board.

The concept of a "safe check", i.e. a check delivered by a piece that cannot result in a trivial capture by an enemy king, was first used by Taylor in 1876. Taylor reasoned that the probability associated with delivering a safe check on an empty board featuring just the piece in question and the enemy king should be proportional to its strength.

Let's take a Rook for an example. Ok, place a Rook on a1,a2,a3...a6. It can safely check a king on a8. It cannot safely check a king on a8 when it is on a7, since Kxa7 violates "safe check".

You essentially "sum" these safe checks over the entire board, placing the king on each square, and computing the number of squares on which a rook resides. The ratio of safe checks to total arrangements on an 8x8 board is 1:6 for the Rook.

Bishops get a little messy in the computation (explanation is not too intuitive) but basically varying diagonal lengths as a function of bishop location and king placement make it very recalcitrant to derive. Knight computations are easy, so are the other pieces.

So, I set out to do the same for a board of dimensions Y by Z, not just a square board like Taylor did.

More on the algebra of the Rook computation in the next post...

Hi, Ed! I couldn't help it: When I saw that Fencer had implemented Gothic Chess while I was logged on to BrainKing.com, I just couldn't resist rushing over to be the first to post. :-)

I would like to see how you came up with the formulas for the piece values (i.e. the "safe check probability"). I was able to derive the formulas for the Rook and Knight that you published in your article (Gothic Chess Review, July 2000), but those were the only two formulas you listed. (OK, I'll admit it. I'm a math professor, so I find that sort of thing interesting. :-) )

Hi Juan,

You beat me to the first post! :) Sure, this is a good place to discuss piece values now. Should I show how to rederive them, or just list their values?

Does anyone know if there is a website that is the "home" of Gothic Chess? Or a website maybe with some tips or something?

I just wanted to be the first one to post to the Gothic Chess discussion board! :-) Hopefully, Ed Trice will be along shortly and we can continue the discussion of the value of the Gothic Chess pieces that was started in the Janus Chess discussion board...

Thanks, I'll have to take a closer look at the value of the Archbishop (Janus). In the article, the author makes a cryptic comment about "adding an extra parameter" to account for the fact that the Archbishop can perform an unassisted mate of a lone King. I have to agree with you that my playing experience is that the Archbishop is definitely weaker than the Queen, although I think the Archbishop may be stronger in closed positions (just like the Knight is stronger in closed positions than the Bishop and sometimes even a Rook).

I've already asked for Omega Chess and Gothic Chess and provided Fencer with the URLs explaining the rules, setup, etc. It might still be worth anyone else mentioning it again to him in order to show that there is more than just one interested person. It appears lots of people are playing Janus Chess and I think lots would play Omega and Gothic Chess as well. So, show your support for Omega and Gothic Chess by sending Fencer another note! :-)

Yes,sorry my english is not so good...
I know Omega Chess too and it would be great if we can play it here too. I informed fencer in the last month about "Januschess", but I can not ask him every week for another game.The best would be if you send him setup,rules and images of these games. That is sure the the best way.

I have posted to the 'Features requests' board my request that Fencer implement Gothic Chess (and Omega Chess, another interesting variant). He said something like "Why not?", but it would probably be helpful if you posted a request for Gothic Chess as well.

By the way, when you say the value of the Archbishop was "clear deeper", do you mean that it is "clearly less"?

I tested yesterday the values in Gothic Chess with "Zillions of games" and the result was that the values of the chancellor and the queen was nearly the same and the value of the archbishop was clear deeper. But the application is not very strong.
But I have another idea: Perhaps we can play both Januschess and Gothic Chess. There are only to make some small changes at the rules.
Who asks fencer?

Thanks, you have been most inciteful

OK, the values of the pieces in Gothic Chess (a 10x8 board) are (according to the "safe check" theory) as follows:

Piece....... Value
------....... -----
Knight........2.50
Bishop........3.03
Rook..........5.67
Chancellor...8.17
Archbishop..8.41 (Note: Arcbishop = Janus)
Queen.........8.70

Note: According to this theory, the value of all the pieces goes down from that on an 8x8 chessboard. For example, the value of a Knight, Bishop, Rook, and Queen on an 8x8 board are 3.00, 3.25, 6.00, and 9.25, respectively.

One should, of course, realize that these "values" are, at best, guidelines. The actual values vary according to the position. Just like there are positions where a Knight is better than a Bishop (or even a Rook), there a positions where an Archbishop (Janus) is better than a Queen.

juangrande ... i would be interested to know, from your previous research, the relative values of the rook, knight and bishop when they are on the board with the queen, janus and chancellor. I assume a pawn is 1.0

I have only just started playing and I find this discussion interesting. So far I have found the knight attribute extremely helpful and i think this is because it is the beginning of the game where there are more pieces and fighting is at closer range. I expect, as with the knight, the advantage will dimish slightly as the game progresses and the board becomes more open. I can see why the the Janus and Chancellor are closer in values than the Bishop and Rook as the Janus will not be restricted to one colour diagonals but I am not sure why it should be worth more as the Cancellor has the ability to cover an additional two squares.

OK, I looked up the article on the relative value of the Gothic Chess pieces. Based on a computation of the "probablility of delivering a safe check," the values (on a 10x8 board) of the Queen, Archbishop (Janus), and Chancellor are 8.70, 8.41, and 8.17, respectively. It is interesting to note that this scheme rates the Chancellor the least of the three; however, my experience playing Gothic Chess is that the Chancellor "feels" more powerful than the Archbishop. The most surprising thing is how close the Archbishop and Chancellor are to the Queen in strength.

No that is my own opinion which is based on the worth of knight and bishop (Janus,archbishop) on the one side and the worth of rook and bishop (queen) on the other side. When the worth would be the same if you say ,what would be the worth of the chancellor (rook and knight) in gothic chess? It is sure a little bit stronger then a Janus and then a little stronger as a queen? i think no

Interesting. Is this value of the Janus published anywhere or is it based on your experience? In Gothic Chess, giving up a Queen for an Archbishop (the same as a Janus) is called the "Gothic Exchange" and is considered less of a sacrifice than the standard Exchange (giving up a Rook for a Bishop).

I think the Janus has a little less worth as a queen, I would say the value is 7,5-8,0. But the Janus has one special advantage: He can alone checkmate the king in the corner and in the middle of the game you have very interesting tactical possibilities

The Janus is worth almost a Queen, according to an analysis by Ed Trice, the developer of Gothic Chess (which has a piece that moves the same as a Janus but is called an Archbishop). I think a good estimate is about 8.5 for a Janus.

Does anyone know what the acceped relative value of the Janus piece is (given that pawn=1 / knight=3 / bishop=3 / rook=5 / queen=9 or 10)?