CHESMAYNE
Values
Value of the MPs/mps - Level-1 - Traditional chess
This depends on various factors but a rough guesstimate is as follows…
01 QU1 = 9
points
02 RO1 or
RO2 = 5 points
03 BS1 or
BS2 = 3 points
04 KT1 or
KT2 = 3 points
05 PA = 1
point
06 GU = 2
points
An approximate value can be given to each MP/mp
apart from the KI. In certain cases the value of a MP/mp can
dramatically increase or decrease, according to what the MP/mp can
achieve. Together, for example, a BS and a KT are more useful than a RO and PA combination, even though the
value of each pair is six PAs. These
rough values can assist you to work out when you should or should not capture your opponent’s
MPs/mps, and whether you should worry about your own MPs/mps being
captured. The values can also help in
assessing which side is ahead - with careful play the side with the highest
total MP/mp value, or the most material,
will usually (but not always) win
the game. The value of the MPs/mps is forever varying,
their relative values dependent on the mutations of position. Their exact value is constantly modified by
the circumstances of time, position and opportunity. Only experience will enable you to determine
accurately which to give up and which to keep in a particular situation.
Although QU1 is the most powerful chess
MP, the KI is the most important and
must be guarded very carefully at all times.
The KI cannot be valued because if you lose him the game is lost. It does not matter how many MPs/mps you have
if you are going to lose your KI.
01 Fair exchange: if you lose a MP
of the same value as the MP you capture.
02 Exchange advantage: if you lose a
MP of less value than the MP you capture.
03 Sacrifice: if
you lose a MP of more value than the MP you capture.
MP/mp values are a useful average guide in play
but you also have to bear in mind other factors. A PA on rank-7 that is about to be
promoted/enrobed, is worth
more than one point. To exchange BS1
for BS1 is not good if your BS
is more mobile than your
opponent’s.
Value of
the MPs/mps :L01
Over 20% of moves in chess are captures. So it is vital to have an idea of the
average values of the various units of force.
Each MP/mp has an average value.
For QU1, RO1 and RO2 and BS1 and BS2, it is proportionate to the average
number of cells
controlled. An unobstructed RO always
controls 14, but a BSs range varies from 13 to 7, and averages just under
9. On this basis a RO is worth at
least 1.5 BSs, and in practice this figure holds. Similar calculation would rate a KT as much
weaker than a BS, but special factors favour the KT so much that its true
average value approximates to a BSs. Factors
favouring the KT are that it cannot be obstructed, and it has access to all the
cells, a BS only to 32. In freak
positions the average values can be temporarily quite false, but normally they
are a good guide. The unit is always
taken as an average PA ie, a PA that has no special advantage, like being
passed (:pa-PA), or
part of a checkmating net. Average values are,
Value of
the RO
Winning a RO for a KT or BS is called ‘winning
the exchange’. The exchange is worth on
average, nearly two PAs. Two BSs or two
KTs equal a RO and two PAs. Two KTs or
BSs are worth only a RO and one PA if the other MPs have been exchanged. ROs like open spaces and are therefore
strongest in the endgame when
the board is clear of obstruction. Two ROs may be exchanged for BS1, BS2 and KT
or, KT1, KT2 and BS.
Value of
QU1
Before the endgame QU1 is worth two ROs, but
in the endgame the two ROs are nearly always worth a PA more, unless the KI is
exposed to +CH. Another approximate equivalent of QU1 is
three KTs or three BSs or two KTs and one BS.
They are almost always at least equal to QU1. Two BSs and a KT are usually superior. RO and BS or KT plus two PAs are almost
always superior to QU1. Exceptions to
these valuations occur where the opposing KI is very exposed so that QU1 can
organize many +CHs, giving herself virtually several successive moves while
your opponent is powerless to make useful replies. QU1 may be exchanged for two ROs and a PA,
but towards the end of a game she is not as valuable as two ROs.
About His
Majesty - the KI
Since the fate of the game hangs on the KI, he cannot
be given a precise numerical value. However,
when so many MPs/mps have been exchanged that checkmate is no longer a
serious danger, he may and indeed must be used as a fighting MP. As such the KI rates well below a RO but
above a BS or KT. When most of the MPs
have been exchanged, including the QUs and at least one pair of ROs, your KI
ceases to need shelter of PAs in a corner.
Use him as a marauder in your enemies lines.
01 BS or KT = 3 PAs (3.5 in the early stages).
02 RO = value of KT or BS plus 1.5 to 2.0 PAs.
03 QU1 = 2 ROs, or 3 KTs, or 3 BSs, or RO, KT or BS and 1.5 PAs.
04 Summary: KT, BS = 3 to 3.5. RO
= 5. QU = 9 to 10.
Some MPs/mps are, of course, more valuable than
others. Often the value of a MP/mp
depends on its position on the board and what it can do in this position. A BS stuck behind mps, or in a corner, is
not as powerful as a central KT with more freedom of movement. When exchanging MPs/mps, there should be some
evaluation. Remember: all PAs (mps) are of equal value,
but some are more equal than others.
The values of the traditional chess
MPs/mps - RO, BS, KT, QU and PA have been found by experience to be
approximately proportional to 1, 3, 3, 5 and 9. A KI is worth about 4 in the endgame. These values vary with the position and with
the number of MPs/mps on the board. For
example, two KTs are worth less than a RO when the only other MPs/mps on the
board are two KIs, in so far as two KTs and a KI cannot force checkmate against
a lone KI. Two KTs are about the equal
of six PAs on rank-2 even
when the KIs are removed. A theoretical
attempt to evaluate the MPs/mps was made by H.M. Taylor in 1876. The value of a MP/mp is taken as
proportional to the average number of cells controlled, averaged
over all cellular positions of the MP/mp on the board. This argument leads to the relative values
of KT, BS, RO and QU1 - 3, 5, 8 and 13 respectively.
Coxter and Taylor went on to
modify this argument by asking instead for the probability of ‘safety’ giving
check, that is, without being ‘en prise’ to the KI, if the
MP/mp and KI are both placed on the board at random.
From http://www.chess-poster.com
The
numerical value of the chess pieces accepted by most chess players is as
follows:
KI........... Infinite - you lose the game if you lose him!
QU........ 9 points
RO.......... 5 points
BS....... 3 points - Bobby Fischer values him at 3.25 points
KT........ 3 points
PA......... 1 point
Of course those values are relative to their strategic position on the board
during the game, i.e., a passed PA on the 6th or 7th rank has a possible potential value of 9 points or so.
A win is 1 point, a loss 0 and a draw ½ point for each player.
Ideal
Values and Practical Values - part 01
by Ralph Betza
My research into the values of chess
pieces was conducted with the specific end in view of making it possible to
construct the game of Chess With
Different Armies, although of course the research has also contributed to
many other interesting chess variants.
In this research, I found that the basic pieces formed from a single
type of movement, that is, the Alfil, Dabbabah,
Ferz,
and Wazir,
had very different values when considered as pieces in their own right - but
when one of these pieces was added to a Knight, the combination of the two was
equal for all practical purposes. (The
NA is as strong as the ND which is as strong as the NW or the NF, even though
the F by itself is worth nearly twice the A by itself.)
In fact, it seems that each piece has two values. One value is the ‘practical value’ of that
particular piece, considering all its specific weaknesses and strengths; and
the other value is the ‘ideal value’ of that piece, the ‘abstract value’ that
the piece has when combining the moves of two pieces masks the weaknesses of
both.
I devoted a great many words to the question of practical values, and
came to the conclusion that although I could not really solve the problem I
could at least develop some useful guidelines; however, I largely ignored the
question of ideal values.
As it turns out, ideal values of simple one-step pieces are absurdly
simple: however many different moves it can make, that’s its ideal value. An Alfil moves to 4 different squares, so it
is ideally worth half as much as a Knight, which moves to 8. A piece that combines the moves of Alfil and
Dabaaba moves to 8 different squares, so it has the same ideal value as the
Knight.
Although this is absurdly simple, I believe it to be a Truth; and
equally, the ideal value of a piece combining many moves is the most important
component of its practical value.
For example, the practical value of the AD is much less than the
practical value of a Knight, and it is obvious that the main reason for the
weakness of the AD is caused by its extreme colorboundness (it can only go to
squares of one color, and what’s worse, it can only go to half the squares of one
color). If we mask that weakness by
adding a W, we get the WAD, which is as strong as a NW for all practical
purposes.
“For all practical purposes”
is vague because A piece is as
strong as the hand that holds it. A
grandmaster who is thoroughly familiar with both pieces might find that one or
the other has a winning advantage; but a mere master who has only a nodding
acquaintance with them does in fact find them to be roughly equal.
In fact, experience shows that a NW or a WAD is in practice roughly as
strong as a Rook, and by extension the NWAD or NWB (adding something the value
of a minor piece to something the value of a Rook) must be roughly as strong as
a Queen.
When I realized that this simplification could be made to work, I was
pleased and excited - think about how regular it is! Take a pile of piece components each of
which is ideally worth half a Knight (the W, F, A, and D of course, but also
the Crab,
the Barc,
the narrow half Knight, and the wide half Knight), and if you combine any two
of them you have something ideally worth as much as a Knight or Bishop; combine
any three of them, you get a Rook’s value; combine five, and you’ve got a
Queen.
How simple, how regular, how
symmetrical!
Not only that, but if you combine four of these units, it’s an Archbishop
(Bishop plus Knight, a piece seen in many chess variants); combine seven
(forget that the choices I listed only go up to 6) and it’s an Amazon;
combine six, and it’s a piece of a value that’s rarely been used (worth as much
as a piece combining the Rook and the Knightrider).
How beautiful, simple, regular, and symmetrical! The several common values of chess pieces
are reduced to a simple quantity, take two, take three, take five.
The ideal values at least are thus reduced. The practical values can still raise
complicated questions. On the other
hand, it is easy enough to choose combinations of choices that mask the
weaknesses of the components and bring a piece approximately to its ideal value
in practice - for example, adding W or Crab to a colorbound piece nearly always
does the trick.
“Nearly always does the trick”
should be exciting to all the chess variant inventors out there. It means that you can design chess variants
in which the players have different armies, using pieces that have never been
seen before, and although you will still want to playtest it a bit you can
expect to have a good chance of getting it right the first time.
For example, suppose you invent a new piece called the Crabbish, which
combines the powers of Bishop and Crab.
You know from the start that its ideal value is exactly a Rook, and you
also know that its practical value is sure to be very close to that of a
Rook. You can give one player a Rook,
and give the other player a Crabbish, and you can be fairly confident that you
haven’t created a major problem with your game’s balance. (In practice, K plus R versus K is an easy
win and K + Crabbish versus K is certainly difficult and perhaps impossible;
but if the player with the Crabbish has this “can-checkmate” advantage with
some other pair of pieces, for example WD versus N, it should even out.)
There is more to be said on this subject, but this seems a good place to
stop, and allow both the author and the reader to think about what has been
said so far.
Click
here
for the next article in this series.
Written
by Ralph Betza.
Ideal Values and Practical Values - part 2
Atomic Pieces and Subatomic Movements
By Ralph Betza
Alfil,
Dabbabah,
Ferz,
and Wazir,
and Knight
are the fundamental geometrical units of Chess; each moves and captures one
square in all instances of type of movement.
They are basic pieces formed from a single type of movement.
I have said they were the atoms of Chess, but in fact they can be
further divided.
The Ferz, for
example, can move in four directions and can capture in four directions; and it
is possible that a piece could be composed that would use just a few of the
eight movements that make up a Ferz.
Indeed, such a piece exists, and Philidor said it was the
soul of Chess.
If a combination of any two atoms gives a new piece whose ideal value is
the same as the ideal value of the Knight, then wouldn’t you expect that the
combination of sixteen movements would create a new piece with Knightly
ideals?
In fact, I think it does create such a piece. However, there are several reasons why such
pieces may have practical values much lower than their ideal values. For example, a piece with sixteen different
non-capturing movements will be nearly worthless (see my article about the Black Ghost),
while a piece with sixteen captures but which cannot move without capturing is
less bad, but not nearly worth a Knight in practice; also, pieces that have
many non-capturing movements can sometimes suffer from being blocked by enemy
pieces, pieces with many more forward movements than rearward are too strong
(because you start the game with foe in front; Hans Bodlaender has pointed out
that much of this advantage dissipates in the endgame, but I feel that by then
the excessive forwardness should have won the game) and pieces with too many
retreats are too weak.
Even if the forward and rearward movements and captures are balanced, so
that the piece has a practical value equivalent to its real value, many
possible pieces are too much of a hodgepodge to be interesting. For example, a piece that moves Northwest as
a narrow Knight, captures Northwest as a wide Knight, moves and captures
Northeast as an Alfil,
moves South in retreat as W or D, captures Southeast as a Ferz, and captures
Southwest as a Ferz, may have the practical value of half a Knight, but who
would want to have such a piece in their game?
Unless, of course, the game was based on having such awful pieces; for
example ‘Subatomic Chaos Chess’, in which you make a legal move and then
randomize one piece, be it friend or foe, but not K or P (otherwise K versus K
might be a win for a randomly-chosen player).
“Randomize” means you throw
the dice to choose 16 subatomic movements for a N, or for a rider such as R, Q,
or B, you randomly choose the right number of moves or captures in the
directions W, F, and N. (The Queen
could become a Nightrider.) You can’t randomize the piece your opponent
just randed. You must randomize if you
can, no penalty if you cannot, and if randomizing an enemy piece puts you in
check, you lose. Basically, the rules of
Avalanche
Chess are applied.
Randomizing the moves by hand could be tedious. With a computer
moderator, this might be an interesting game.
Thousands of new pieces can be created by combining the subatomic
movements, and even if most of them are bad in some way, the remaining small
portion must contain dozens, even hundreds, of possible new pieces that would
be useful in Chess variants. I cannot
make such a statement without choosing an example, so consider the left-handed Commoner,
right-handed Waffle:
it moves and captures as a W in all directions, to the left as F and to the
right as A. You would start the game
with one sinister and one dexter of this charming new piece whose practical
value is very probably close to the Knight’s.
Of course, the Rook, Bishop, and Nightrider can also be decomposed in
similar fashion, and used to form combination pieces whose ideal values can be
described simply by adding up the appropriate fractions of the known values of
these pieces. A Knight that can also
move left and capture right as a Bishop should be worth a Rook; a piece moving
and capturing West or South as Rook, but NNW, NNE, ENE, and ESE as Nightrider
should also be worth a Rook; and both pieces should be fairly easy to learn to
use.
Now I have divided the ideals into their smallest possible parts. Next article, I will run with the great
unsolved mystery of piece values.
Click
here
for the previous article in this series.
Click
here
for the next article in this series.
Written
by Ralph Betza.
Ideal Values and Practical Values - part 3
The Rider Problem
By Ralph Betza
The Rook is technically known
as a Wazir-Rider: it makes a Wazir
move, and if it lands on an empty square it may make another in the same
direction, repeating this process any number of times. Of course, the Bishop is a Ferz-Rider.
The ideal values of Ferz and
Wazir are the same, but the well-known practical values of the Rook and Bishop
are quite different. Why?
The ideal values of the R and B are presumed to be very close to their
practical values, and so the Rider Puzzle is very much in the scope of a
discussion of ideal values.
I have mentioned more than once that there is a formula for the
probability that a given jump is on the board:
Consider a
one-step move of displacement x and y (for
example a Knight move has x = 1 and y = 2 (or x = 2 and y = -1 and so on for
all combinations, but we call that a (1,2) jump)) being made on a board
of dimensions w and h (the normal
chessboard has w = 8 and h = 8) the answer is ((w - x) times (h - y))
divided by (w times h).
And for a Knight the result
is (7 * 6) / 64, but there are 8 directions in which a Knight can move, and so
we multiply the result by 8, and it comes out to 5.25 which is exactly the
average number of moves a Knight has when you put it on every square and count
them up. So there.
A Rook makes an (0,1) jump, then if it landed on an empty square it may
continue to (0,2) and so on. Therefore
its average mobility is
the probability that (0,1) is on the board plus the product of the probability
that a square is empty and the probability that (0,2) is on the board, and so
on. The probability that a square is
empty varies (gets larger as the game goes on), so there isn’t one perfect number
for the average mobility of a rider; and although average mobility is a very
important part of piece values, I can’t find a reliable way to calculate one
from the other.
It would be nice to know both the ideal and the practical values of rider
pieces. Even the Rook and Bishop, whose practical values are fairly well-known,
have unknown ideal values. I assume
that the ideal value of the R is roughly equal to its practical value, and that
the ideal value of the Bishop is a bit larger than its practical value; one
possible clue is that the Queen is worth a notable amount more than the
separate R and B, but this seems to be mostly because pieces that concentrate
great value are as a general rule worth more than their separate component
pieces (more forking power).
The Chancellor
is roughly equivalent to the Queen even though the ideal value of N is
presumably less than Bishop: the Bishop is colorbound and its practical value
is ever so slightly more than a Knight, combining it with R removes the
colorboundness, and therefore is a classical case of “combining pieces to mask
their weaknesses and thus allow their practical values to be fully expressed”;
and therefore one might expect the Q to be worth notably more than the
Chancellor.
One hypothesis about why the Chancellor does so well is that the R has a
weakness that is masked when N is added to form Chancellor. This weakness would be its relative slowness
and difficulty of development, and perhaps its lack of forwardness (it has only
one forwards direction).
Digression: the endgame with WKa1, BKg7, BPf7 is drawn if W has a Q,
but won if W has a Chancellor. You
should work it out for yourself because it’s interesting.
The Nightrider
is another special case. It moves in
twice as many directions as the Rook, but covers two squares with each step and
therefore cannot take many steps in one direction before being stopped by the
edge of the board. Despite this, its
practical value is very much equal to the Rook - but one might expect the ideal
value of the R to be larger than the NN’s.
The Dabbabahrider moves in (0,2) increments as opposed to the Rook’s
(0,1) steps. As a piece by itself, it
is much weaker than a Knight, the main reason being that it is colorbound times
colorbound - it can visit only one fourth of all squares on the board. The Alfilrider is even worse, and can see
only one eighth of all squares. Because of this extreme limitation, we have the
interesting case where the AD (Alfil plus
Dabbabah)
has the same ideal value as the Knight but is much weaker in practice, while
the AADD (Alfilrider plus Dabbabahrider) has an ideal value which is unknown
but which must be appreciably larger than Knight - but the practical value of
AADD seems to be a bit less than a Knight.
In order to use these atomic movements in combination with others and
thus define new pieces, one would like to know their ideal values.
One way of looking at the rider-value is that the Wazir-rider (the Rook)
seems to be worth 3 times the Wazir, while the Ferzrider (the Bishop) is only
worth twice as much as a Ferz; and remember, the ideal values of Wazir and Ferz
are the same. (I really want to find a
formula using the inverse of the geometrical distance of one step [1], but
can’t find any convincing reason.) The
Nightrider is worth a mere 1.5 times as much as a Knight. The multiplier for Dabbabahrider should be
somewhere between 1.5 and 2, while the multiplier for the Alfilrider is
probably less than 1.5 (and by definition must be greater than 1.0).
If the multiplier for the DD is 1.75, exactly between the two proposed
limits, the difference between D and DD is worth a Pawn; and if the AA is worth
1.25 times the A, the difference between the two pieces is just one the
“quantum of advantage”, the smallest difference in values that you can
notice.
The above two paragraphs give a better idea of the values of AA and DD
than we ever had before, but aren’t very exact nor probably very accurate.
I can’t solve the Rider
Puzzle. This is depressing. Let’s have some fun instead.
The Avian Airforce
Consider, for example, the Falcon,
FAA, Ferz plus Alfilrider. The FA by
itself has an ideal value equivalent to N or B, and in practice is worth
perhaps one quantum of advantage less than N or B. Making the A into an AA should add the
missing quantum, and (because of concentrated power is worth more) maybe even
more. Therefore I expect the FAA to be fully equal or even slightly better than
N or B.
Consider also the Wader,
WDD, Wazir plus Dabbabahrider. The WD
by itself is a good match for the N, weaker in the opening but stronger in the
endgame. The WDD must clearly be
slightly weaker than the R, but it’s definitely in the “major piece” range so
that the levelling effect should help the WDD by pulling its value up towards
R.
Combined, we get the Flying
Kingfisher, WFAADD, which must be worth about as much as a Queen, how
much more or less who can guess?
That’s nearly a whole army. What
to use for the Knight? Wazir plus long
crab is tempting as “more of the same”, but development can get too awkward. Commoner
would provide balance - this army is a little bit weak but has so much early
mobility that it should be playable; using Commoners as Knights would give it
some endgame punch - but so common. I
want something brand new, something subatomic in fact...
Thus, the Darter: moves
forward as narrow Knight, all directions as Wazir, and retreats as -
Alfilrider!! The Alfilrider retreat
increases its value so it’s a tiny bit stronger than a N, but the army as a
whole needs extra strength and so it’s okay.
The riding retreat is not nearly so strong as a riding advance, and
since long diagonal retreats are the moves that Grandmasters most frequently
overlook, there’s a certain charm to the choice.
The Avian Air Force would
be the army’s name, and of course it’s classed as an experimental army.
·
The immediate threats of 1.
WFAADDd1-b3 or WFAADDd1-d3 (wanting to win Ra8 by checking at a4 or e4) are
easily answered by the The
Fabulous FIDE army.
- The Clobberers
must use their alternate setup when playing against the Avians, else a7/h7
are hanging.
- The The
Nutty Knights may suffer from their undefended g7.
- If the Avians fail to gain sufficient advantage
from their early mobility, they will lose in the endgame...
You can see how research into values leads directly to creation of new
chess variants. On the other hand,
using the Avians can be construed as research because practical experience with
the Faalcon and Wader and Kingfisher will make it possible to refine the blind
guesses about the ideal values of the AA and DD!
In the next article, I will ferociously attempt to establish ideal
values for some other types of pieces, but will probably fail.
[1]: An
interesting point is that the practical values of R and B in ‘Cylindrical
Chess’ are thought to be roughly equal by players who have much experience in
the game.
Click
here
for the previous article in this series.
Click
here
for the next article in this series.
Written
by Ralph Betza.
Ideal Values and Practical Values - part 5
Modified Jumpers
By Ralph Betza
The “lame” jumper is a piece
that cannot leap over an intervening square if that square is blocked. For example, a lame Alfil can
move from f1 to d3 only if e2 is empty.
The magic number is clearly applicable here, and it says that the lame
Alfil is worth somewhere around 0.66 to 0.7 times as much as a normal (leaping)
Alfil.
By the same reasoning, the lame H (the non-leaping version of the (0,3)
jumper) is worth 0.49 (or 0.44) standard atoms because it needs both f2 and f3
to be empty in order for it to go from f1 to f4.
A Knight-valued piece can be
made combining:
1.
Lame H with
2.
Vertical Wazir
(forwards and backwards but not sideways) with
3.
Narrow Knight (from e4 to
d6, f6, d2, or f2 but unable to go from e4 to c3, c5, g3, or g5).
and this piece
would be reasonably well-rounded and pleasant to use, with its opening value
enhanced by its leaping move and its endgame value enhanced by its long
non-leaping move. Its endgame value
would be greater than N, but in the context of an army that is otherwise weak
in endgame value that would be a good thing.
Another type of lame jumper can be seen in the zF2, a Crooked
Bishop limited to short moves. In
effect, it makes either a Ferz move
or a lame Dabbabah
move, but instead of the D jumping over the square in front of it (a normal
lame D could not go from f1 to f2 if f2 was occupied), it jumps over either of
the diagonal squares (the zF2 can go from f1 to f3 if e2 is empty or if g2 is
empty). The probability that both of
two squares are occupied ranges from 0.09 to 0.16 depending on the magic value,
and so the crooked lame Dabbabah is worth 0.91 to 0.84 times as much as a
normal D - not enough difference to make a difference. The same math would apply to an H that was
allowed to jump over zero pieces or one piece, but not two.
The Cannon
from Xiang Qi is a
special case of lame jumper. When it captures,
it is allowed to jump over one piece, but not to jump over zero pieces. Simple subtraction shows that if a normal
lame H has 0.36 to 0.49 as much mobility as a leaping H, and if an H jumping
either zero or one piece has .84 to .91, then the cannon H (gH, jumping one but
not zero nor two pieces) must be 0.48 to 0.42 as much as an H; but the trick is
that the normal lame H increases in value as the board empties out but the
cannon H loses value when there are fewer pieces for it to jump over.
The traditional Xiang Qi Cannon, which moves as R but captures as pR,
provides balance and is an excellent combination because as the board empties
out it becomes easier to move the piece but harder to capture with it (its R
component benefits from the open board but its pR component suffers from the
same cause).
Note that in Xiang Qi both Kings are confined to their castles and
therefore the Cannon can always be useful, using its own K as a screen and the
enemy King’s defenders as a target.
This means that the imbalance of movement and capture power that would
afflict the Xiang Qi Cannon if used in a FIDE setup does not hurt it nearly as
much in an Xiang Qi setup.
The simplifications presented in the earlier articles of this series
allow us to say that the ability to move like a Rook (but not to capture in
that manner) is worth half as much as a Rook, as long as the capturing power of
the final piece is above some minimum theshold value. The threshold is unknown, which is
unfortunate, but in the late endgame when very few pieces are on the board the
power to capture as pR certainly falls below that threshold. See this page
for details.
In the initial position, the average mobility of pR is two-thirds that
of R, around the magic number it is half, and late in the game it falls as low
as one sixth. I’m willing to take
“half” as a guess, so that the combination of mRcpR would be theoretically
worth 0.75 as much as a Rook, but with the warning that you really need to
trade it off before the endgame. I
follow Rudolph Spielmann’s belief that a minor piece is worth two thirds of a
Rook (not the traditional 0.6), and I have won tournament games against strong
players based on this belief in a difference of estimated values amounting to a
mere 0.0666 of a Rook; because the mRcpR’s estimated value is 0.08 of a Rook
more than a minor piece (that would be 0.4 Pawns, an advantage large enough to
feel), according to this guess at its value having a pair of standard Xiang Qi
Cannons against a pair of Knights should be enough to win on an 8 x 8 board
with the standard arrangement of 32 pieces, and all we need is a bit of
playtesting to find out if the guess is correct.
Actually, it’s not so simple.
Take the standard FIDE army, replace the Knights with mRcpR Cannons, and
now stare at the board for a while and try to figure out how to develop your
pieces! Actually, even replacing the
Knights or Bishops with Rooks isn’t that easy to win, so doing it with mere
Cannons would perhaps cancel completely any material advantage.
An alternative would be to replace B with HFD and R with mRcpR; although
this adds an advantage of rapid jumping development to the presumed material
edge, I think that one could get a feel for whether or not the Xiang Qi Cannon
is really worth that much more than a minor piece by playing several games of
this; or of course one could simply replace the R4 of the Remarkable Rookies
with mRcpR Cannons...
Part of the problem with cannon values is that the FIDE setup is dense
and crowded. In Xiang Qi, when a
Cannon aims into the enemy position there is room to interpose or to step
aside, but against the FIDE setup, a Cannon attack may be instantly fatal - for
example, if the piece on a1 both moves and captures as cannon-Rook, then 1
pRa1-a3 threatens to capture Ra8, and after 1...Nb8-a6, simply 2 Nb1-c3 e7-e6 3
pRa1-e3+ Bf8-e7 4 pRh1-h3 Ng8-h6 5 pRh3-d3 wins the Queen. Thus, there is always danger with Cannon
pieces, and you need to be very careful when adding one to a game.
My new idea that balance makes it possible to get better estimates of
piece values tells me that if a piece has sufficient non-Cannon movement,
preferably movement that increases in value as the game goes on, it ought to be
possible to make a good guess about its value based on its average mobility
using the magic number for emptiness of the board.
The idea of balance also tells me that it’s not the best idea to make up
themed armies like the Clobberers or the Avians; but these armies are perhaps
more fun to design and more fun to play than balanced armies would be, and
that’s a value in its own right.
The Spacious Cannoneers
Common sense also tells me
that making up an army whose power is based mostly on the combination of a new
and untested estimate of the values of Cannon pieces plus a new and untested
kind of piece is risky; but what the heck, if the Spacious Cannoneers turn out
to be too weak or too strong when used in different-army games, at least one
can always enjoy games where both sides use the same army, so let's do it.
The Spacious Cannoneers R = the Mortar and the
Howitzer
The Spacious Cannoneer Rook comes in two flavors, and I like the idea of
having both in the same game.
Both kinds of Rook move and capture as Wazir or as Spacious
Rook. (The Wazir power is useful only
when the Spacious Rook power cannot make a one-square move.)
To this, the Mortar adds the power of capturing (but not moving) as a
Rookwise Cannon, while the Howitzer adds the power of moving (but not
capturing) as a Rookwise Cannon.
It is recommended to start with Mortar on a1 or h8, Howitzer on h1 or
a8; Black gets the one that attacks the enemy Kingside position as compensation
for White's first-move advantage. In
games with different armies, perhaps only the Howitzer should be used.
Valuewise, the Spacious Rook should be worth two thirds of a R, half the
power of a Cannon Rook should be worth one quarter of a R, and the small amount
added by the Wazir move should be worth one ninth of a R. This adds up to a bit
more than a Rook, not enough to be significant (the error in the estimate of
value is likely to be much larger than that!).
The Spacious Cannoneers B == the Carronade
The Spacious Cannoneer
Bishop is named the Carronade and moves and captures as Spacious Bishop or as
Cannon Bishop.
Valuewise, two thirds plus
one third, nothing could be more simple.
Note that 1. Carronade
f1-b5+ is legal but not a good move.
The Spacious Cannoneers Q == Big Bertha
The Spacious Cannoneer Queen
is named Big Bertha and combines the powers of Howitzer and Carronade.
Note that combining Mortar with Carronade would allow 1. BigB d1-h5+ or
1. BigB d1-a4+ not only winning a R, but also with unstoppable 2. BigbB-e4 (or
to e5) checkmate.
The Spacious Cannoneers N == the Napoleon
The Napoleon is an equine piece that is neither Spacious nor
Cannonistic, but is named the Napoleon after the horse-drawn field artillery of
Napoleonic times.
I have chosen the fbNW - narrow Knight plus Wazir - simply because it is
a good Knight substitute that I have often mentioned but have never used in any
game.
The Spacious Cannoneers == Summary
This army is highly experimental and could be much too strong or much
too weak to use against other standard armies.
This army is quite exotic and takes some getting used to. Players who haven’t played enough Xiang Qi
to get accustomed to cannon moves may well be baffled by a game that combines
two exotic powers at once. Players who
know the Cannon well will still be startled by the Spacious pieces.
In the late endgame, the long-distance pieces lose very little by being
Spacious and gain very little by being Cannonate; but the small tactical
details can be interesting, when the difference between a Bishop and a
Carronade suddenly makes itself felt.
In the early game, Spaciousness should frustrate many moves you want to
make, while Cannonization should give you pleasant choices to make up for that.
The Spacious Cannoneers == Sample Game
Let us suppose W has the Fabulous FIDE army, and Black has the Spacious
Cannoneers with Mortar on h8, Howitzer a8.
1. e4 e5 2. Nf3 d7-d5
The fbNW is a good piece in
general, but can’t defend e5.
2...Carronade
c8-g4 might be good. It pins the Nf3
with Spacious power and attacks the Qd1 with Cannon power, and after the forced
Bf1-e2, both e2 and f3 are pinned by the Cannon power, though the Nf3 is not
attacked.
2...d7-d6 does
not block the Carronade at f8 from developing to c5, however it does block it
from developing to e7!
The Pe5 is
halfway defended, because 3 Nxe5 BigB d8-a5+ 4 d2-d4 (f8 is occupied, so Spacious
capture on e1 is illegal) looks mighty risky.
But wait a minute! 1. e4?, BigB d8-h4+ and wins the Pe4.
1.e2-e4? is
just a bad move, because 1. d2-d4 BigBd8-h4+? 2. g2-g3 BigBh4-e4 3. Bf1-g2!
defends, develops, and drives back the attack with loss of time. On the other hand 1 d2-d4 allows Carronade
f8-b4 checkmate.
3.Bf1-g2 is
picturesque, don’t you think? But in
fact the Rh1 wasn’t attacked because a8 is occupied.
Hmmm, 1 e2-e4 BigBd8-h4+ 2 g2-g3 BigBh4xe4 (not check!) 3 Nb1-c3
Carronade f8-b4 (pin) 4 Bf1-g2 BigB e4-g6! attacks g2, White doesn’t have
enough compensation for the Pawn.
Instead 4.a2-a3 Carr b4-a5 (now b2-b4 allows Carr a5xc3), 5 Qd1-e2 BigB
e4xc2, not enough compensation.
Therefore 1. Ng1-f3 is going to be a popular opening move. 1 Nc3 is also good, or 1.g2-g3, and enough
other choices that the opening is playable.
If White has the Spacious Cannoneers with two Howitzers and Black has
the Clobberers, 1. BigB d1-h5+? g7-g6, e5 is defended by BDh8, h7 can’t be
captured because h8 occupied, W must retreat in shame; or if Black had the
FIDEs, 1. BigB-h5+ g6 2. BigB-e5 makes no threat, is simply a bad move.
As I said, Cannon powers can be dangerous. It appears as if there is danger here, but
that the opposing army can, with a bit of care, always survive and even profit
from attempts at early raids.
In addition, this little bit of analysis gives me more faith in the
proposition that the weakness of Spaciousness balances the strengths of
artillery.
I’ve always wanted to make up a variant where one side had the FIDE army
but the other side had an army with Cannon pieces. Well, not really always, only since 1977 or
so, close enough.
Click
here
for the previous article in this series.
Click
here
for the next article in this series.
Written
by Ralph Betza.
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 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 8
x 8 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 8 x 8 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 Balance.
[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.