I hope very much Roger does not delete this thread, if I am the only poster in it, as it might be the case that a few people will read it at some point, even if not just now.
In any case, even if Roger deletes it, no big loss, as it will free more space for topics on checkers, legal issues and psychological investigations.
Chain, duo, connected
Moderators: hgm, Rebel, chrisw
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Lyudmil Tsvetkov
- Posts: 6052
- Joined: Tue Jun 12, 2012 12:41 pm
Re: Chain, duo, connected
Well, a chain pawn is any pawn that is part of a chain, but here we will mainly refer to chain pawns as to defended pawns, as the base/undefended pawn of the chain is much less significant and actually only due a penalty.
Duos are of course, according to Kmoch's definition, 2 own pawns next to each other on the same rank on adjacent files. A duo pawn is one of those pawns.
What concerns connected pawns, here I will refer mainly to SF's definition of connected pawns, however imprecise it might be, that considers both chain/defended pawns and duos as connected pawns, giving the same bonus.
[d]6k1/8/8/8/3P1PP1/2P5/8/6K1 w - - 0 1
above, d4 is a chain/defended pawn
f4 and g4 are a duo as a tandem, and a duo pawn each of them
SF scores both d4, and f4 and g4 in exactly the same way, as connected pawns. In practice, this works fine, as when the pawns move forward, they support each other by alternating more and less advanced duos and chain pawns, so a continuity is ensured.
However, scientifically, there is a distinction in terms of the size of the bonus between the d4 chain/defended pawn and the duo pawns.
The chain/defended pawn on the same square is due around 1/3 higher bonus than the respective duo pawn, so basically the primary concept from where connected pawns derive is the defended/chain pawn, although Mr. Kmoch might think otherwise.
Duos are of course, according to Kmoch's definition, 2 own pawns next to each other on the same rank on adjacent files. A duo pawn is one of those pawns.
What concerns connected pawns, here I will refer mainly to SF's definition of connected pawns, however imprecise it might be, that considers both chain/defended pawns and duos as connected pawns, giving the same bonus.
[d]6k1/8/8/8/3P1PP1/2P5/8/6K1 w - - 0 1
above, d4 is a chain/defended pawn
f4 and g4 are a duo as a tandem, and a duo pawn each of them
SF scores both d4, and f4 and g4 in exactly the same way, as connected pawns. In practice, this works fine, as when the pawns move forward, they support each other by alternating more and less advanced duos and chain pawns, so a continuity is ensured.
However, scientifically, there is a distinction in terms of the size of the bonus between the d4 chain/defended pawn and the duo pawns.
The chain/defended pawn on the same square is due around 1/3 higher bonus than the respective duo pawn, so basically the primary concept from where connected pawns derive is the defended/chain pawn, although Mr. Kmoch might think otherwise.
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Lyudmil Tsvetkov
- Posts: 6052
- Joined: Tue Jun 12, 2012 12:41 pm
Apex, chain duo, double duo
Well, an apex is a pawn defended by 2 own pawns.
I never expected this would work in SF, but Mr. Savard found the right implementation.
A chain duo is a defended pawn that is a duo pawn at the same time.
Double duo is a duo pawn that has 2 own pawns on the same rank to the left and right.
[d]6k1/8/2ppp3/6PP/1P3P2/P1P5/8/6K1 w - - 0 1
b4 apex pawn for white, g5 chain duo pawn for white and d6 double duo pawn for black
As you see, b4 is defended by 2 own pawns, a3 and c3, g5 is defended by one own pawn, f4, and is next on the same rank to another own pawn, h5, while d6 black pawn is next to 2 own pawns on the same rank, c6 and e6.
Very similar as those pawns might look to you, there is a big distinction.
Found on the same square, an apex pawn would be much stronger than a chain duo pawn and even stronger than a double duo pawn.
So all are due an additional bonus above the usual connected pawns bonus, but the bonus points should be distinguished very precisely.
If in SF an apex/twice defended pawn succeeded with 1/2 the bonus of connected phalanx opposed in terms of rank, a chain duo pawn might be due somewhere half of that bonus, so 1/4 of connected phalanx opposed, while a double duo pawn would be due somewhere 1/6 of connected phalanx opposed.
How you resolve the discontinuity in the bonus points and how you tune this I have absolutely no clue, but I am more or less certain about the size of the abovementioned bonus points.
In SF they tried to score all twice connected pawns with the same bonus, and even to altogether simplify connected, but the right approach is precisely the other way: specify even more here.
Now, SF and I presume also many other engines completely do not understand this concept, and there is a LOOOT to gain from that, in all kinds of positions, but especially in more closed and semi-closed ones.
If implemented correctly, you would be surprised how much you can gain from that, but you should also have some luck when tuning. In any case, SF misses around 5 of every 6 moves relating to building such twice connected pawns.
I never expected this would work in SF, but Mr. Savard found the right implementation.
A chain duo is a defended pawn that is a duo pawn at the same time.
Double duo is a duo pawn that has 2 own pawns on the same rank to the left and right.
[d]6k1/8/2ppp3/6PP/1P3P2/P1P5/8/6K1 w - - 0 1
b4 apex pawn for white, g5 chain duo pawn for white and d6 double duo pawn for black
As you see, b4 is defended by 2 own pawns, a3 and c3, g5 is defended by one own pawn, f4, and is next on the same rank to another own pawn, h5, while d6 black pawn is next to 2 own pawns on the same rank, c6 and e6.
Very similar as those pawns might look to you, there is a big distinction.
Found on the same square, an apex pawn would be much stronger than a chain duo pawn and even stronger than a double duo pawn.
So all are due an additional bonus above the usual connected pawns bonus, but the bonus points should be distinguished very precisely.
If in SF an apex/twice defended pawn succeeded with 1/2 the bonus of connected phalanx opposed in terms of rank, a chain duo pawn might be due somewhere half of that bonus, so 1/4 of connected phalanx opposed, while a double duo pawn would be due somewhere 1/6 of connected phalanx opposed.
How you resolve the discontinuity in the bonus points and how you tune this I have absolutely no clue, but I am more or less certain about the size of the abovementioned bonus points.
In SF they tried to score all twice connected pawns with the same bonus, and even to altogether simplify connected, but the right approach is precisely the other way: specify even more here.
Now, SF and I presume also many other engines completely do not understand this concept, and there is a LOOOT to gain from that, in all kinds of positions, but especially in more closed and semi-closed ones.
If implemented correctly, you would be surprised how much you can gain from that, but you should also have some luck when tuning. In any case, SF misses around 5 of every 6 moves relating to building such twice connected pawns.
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Lyudmil Tsvetkov
- Posts: 6052
- Joined: Tue Jun 12, 2012 12:41 pm
4,3,2.5,2
Well, this is absolutely essential. You can not do connected pawns without also distinguishing in terms of file.
SF used to have in the past a distinction between d/e, c/f, b/g, a/h files in the proportion of 4-3-3-1, with a/h files scoring just 1, but later this was simplified to equal 3-3-3-3 bonus for all files.
Well, 4-3-3-1 performed worse than 3-3-3-3 mainly because the edge a/h files got very disproportionately low bonus, but an equal bonus for all files is very far from perfect.
For all kind of chain/connected pawns features, you urgently need a distinction.
I think more or less the right weights should be 4-3-2.5-2 from the center towards the edge.
SF used to have in the past a distinction between d/e, c/f, b/g, a/h files in the proportion of 4-3-3-1, with a/h files scoring just 1, but later this was simplified to equal 3-3-3-3 bonus for all files.
Well, 4-3-3-1 performed worse than 3-3-3-3 mainly because the edge a/h files got very disproportionately low bonus, but an equal bonus for all files is very far from perfect.
For all kind of chain/connected pawns features, you urgently need a distinction.
I think more or less the right weights should be 4-3-2.5-2 from the center towards the edge.
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Haikouichthys
- Posts: 38
- Joined: Sun Oct 06, 2013 6:16 am
Re: Chain, duo, connected
SF does not score d4, f4 and g4 the same way. f4 and g4 have an 'unsupported' penalty and a 'phalanx' bonus. Adding all the bonuses and penalties up, we have:Lyudmil Tsvetkov wrote:Well, a chain pawn is any pawn that is part of a chain, but here we will mainly refer to chain pawns as to defended pawns, as the base/undefended pawn of the chain is much less significant and actually only due a penalty.
Duos are of course, according to Kmoch's definition, 2 own pawns next to each other on the same rank on adjacent files. A duo pawn is one of those pawns.
What concerns connected pawns, here I will refer mainly to SF's definition of connected pawns, however imprecise it might be, that considers both chain/defended pawns and duos as connected pawns, giving the same bonus.
[d]6k1/8/8/8/3P1PP1/2P5/8/6K1 w - - 0 1
above, d4 is a chain/defended pawn
f4 and g4 are a duo as a tandem, and a duo pawn each of them
SF scores both d4, and f4 and g4 in exactly the same way, as connected pawns. In practice, this works fine, as when the pawns move forward, they support each other by alternating more and less advanced duos and chain pawns, so a continuity is ensured.
However, scientifically, there is a distinction in terms of the size of the bonus between the d4 chain/defended pawn and the duo pawns.
The chain/defended pawn on the same square is due around 1/3 higher bonus than the respective duo pawn, so basically the primary concept from where connected pawns derive is the defended/chain pawn, although Mr. Kmoch might think otherwise.
(Middlegame, endgame)
d4 : (15, 10)
f4, g4: (29, 23)
So SF actually scores the duo pawns a lot higher than the chain/defended pawn.
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Lyudmil Tsvetkov
- Posts: 6052
- Joined: Tue Jun 12, 2012 12:41 pm
Why long chains do not work?
Well, I will not rest, until I am alive, until I see engines in general and SF specifically succeeding at least partially in scoring long chains.
How much time it needs, it does not matter, years or decades, but I will be repeating this all over again, until I am alive, for the simple reason that long chains could bring you enormous amount of elo.
Why long chains did not so far succeed in SF?
Well, I think one of the main reasons is the equal bonus for all files, a flat file bonus is very bad for long chains, much much worse than a flat file bonus for other connected/chain terms, as a long chain represents a bigger unity and any inaccuracy of the bonus reflects upon all the chain.
SO the obvious cure would be to try to implement first a non-flat file bonus and only then proceed with tuning long chains, as central long chains, with inner pawns passing through the center, d and e files are much more significant than long chains with inner pawns passing through semicentral c and f files, that in turn are way more significant than long chains with inner pawns passing through b and g files.
If you do not make this distinction, you spoil everything with long chains.
Well, in SF long chains scored very well at STC with bonus of 1/16 connected phalanx opposed, so I guess the real bonus should be somewhere around here, or even lower, up to 1/32, however with the added non-flat bonus in terms of files. I guess that might even succeed at LTC.
As a rule, you do not need a big bonus for inner pawns of long chains, on the contrary, the bonus should be very low, just to make a distinction, as the same defended pawns already get their bonus apart, but the low bonus does not in any way mean the term is insignificant. You can have extremely successful patches with terms scoring low bonus points, if the bonus is precisely tuned.
I think the bonus should be around the same for the mg and eg, but it might be the case that the mg value should be higher, as I suppose, or even vice-versa, depending on the specific terms available in an engine, so this also should be investigated when tuning. I think that, in general, the mg long chain bonus is naturally a bit higher.
[d]6k1/5p2/4p3/3p4/3P4/2P5/1P6/6K1 w - - 0 1
the black chain is stronger than the white one, as it passes through the e6 inner pawn on the central e file, while the white chain passes through the c3 inner pawn on the semicentral c file.
[d]6k1/8/2p4P/2Pp2P1/1P2pP2/P4p2/8/6K1 w - - 0 1
extremely powerful black chain passing through 2 inner pawns on the central d and e files, d5 and e4; 2 much weaker smaller white chains passing through the semi-edge b and g files.
The chains for both sides have inner pawns on the 4th and 5th ranks, but the black chain is much more central; also note the efficiency of black resources - 6 white pawns achieve much lesser effect in terms of strength than 4 black pawns.
SO there is also a LOOOT to gain from very long chains with 4,5 and 6 pawns, containing 2,3 and 4 inner pawns instead of just one, but how to try scoring separately very long chains with more than one inner pawns, when engines so far have not succeeded in implementing even bonus for long chains with a single inner pawn?
Again, the only reason I am mentioning this time and again is that there is a lot to gain from that.
How much time it needs, it does not matter, years or decades, but I will be repeating this all over again, until I am alive, for the simple reason that long chains could bring you enormous amount of elo.
Why long chains did not so far succeed in SF?
Well, I think one of the main reasons is the equal bonus for all files, a flat file bonus is very bad for long chains, much much worse than a flat file bonus for other connected/chain terms, as a long chain represents a bigger unity and any inaccuracy of the bonus reflects upon all the chain.
SO the obvious cure would be to try to implement first a non-flat file bonus and only then proceed with tuning long chains, as central long chains, with inner pawns passing through the center, d and e files are much more significant than long chains with inner pawns passing through semicentral c and f files, that in turn are way more significant than long chains with inner pawns passing through b and g files.
If you do not make this distinction, you spoil everything with long chains.
Well, in SF long chains scored very well at STC with bonus of 1/16 connected phalanx opposed, so I guess the real bonus should be somewhere around here, or even lower, up to 1/32, however with the added non-flat bonus in terms of files. I guess that might even succeed at LTC.
As a rule, you do not need a big bonus for inner pawns of long chains, on the contrary, the bonus should be very low, just to make a distinction, as the same defended pawns already get their bonus apart, but the low bonus does not in any way mean the term is insignificant. You can have extremely successful patches with terms scoring low bonus points, if the bonus is precisely tuned.
I think the bonus should be around the same for the mg and eg, but it might be the case that the mg value should be higher, as I suppose, or even vice-versa, depending on the specific terms available in an engine, so this also should be investigated when tuning. I think that, in general, the mg long chain bonus is naturally a bit higher.
[d]6k1/5p2/4p3/3p4/3P4/2P5/1P6/6K1 w - - 0 1
the black chain is stronger than the white one, as it passes through the e6 inner pawn on the central e file, while the white chain passes through the c3 inner pawn on the semicentral c file.
[d]6k1/8/2p4P/2Pp2P1/1P2pP2/P4p2/8/6K1 w - - 0 1
extremely powerful black chain passing through 2 inner pawns on the central d and e files, d5 and e4; 2 much weaker smaller white chains passing through the semi-edge b and g files.
The chains for both sides have inner pawns on the 4th and 5th ranks, but the black chain is much more central; also note the efficiency of black resources - 6 white pawns achieve much lesser effect in terms of strength than 4 black pawns.
SO there is also a LOOOT to gain from very long chains with 4,5 and 6 pawns, containing 2,3 and 4 inner pawns instead of just one, but how to try scoring separately very long chains with more than one inner pawns, when engines so far have not succeeded in implementing even bonus for long chains with a single inner pawn?
Again, the only reason I am mentioning this time and again is that there is a lot to gain from that.
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Lyudmil Tsvetkov
- Posts: 6052
- Joined: Tue Jun 12, 2012 12:41 pm
Re: Chain, duo, connected
Thanks Jarrod.Haikouichthys wrote:SF does not score d4, f4 and g4 the same way. f4 and g4 have an 'unsupported' penalty and a 'phalanx' bonus. Adding all the bonuses and penalties up, we have:Lyudmil Tsvetkov wrote:Well, a chain pawn is any pawn that is part of a chain, but here we will mainly refer to chain pawns as to defended pawns, as the base/undefended pawn of the chain is much less significant and actually only due a penalty.
Duos are of course, according to Kmoch's definition, 2 own pawns next to each other on the same rank on adjacent files. A duo pawn is one of those pawns.
What concerns connected pawns, here I will refer mainly to SF's definition of connected pawns, however imprecise it might be, that considers both chain/defended pawns and duos as connected pawns, giving the same bonus.
[d]6k1/8/8/8/3P1PP1/2P5/8/6K1 w - - 0 1
above, d4 is a chain/defended pawn
f4 and g4 are a duo as a tandem, and a duo pawn each of them
SF scores both d4, and f4 and g4 in exactly the same way, as connected pawns. In practice, this works fine, as when the pawns move forward, they support each other by alternating more and less advanced duos and chain pawns, so a continuity is ensured.
However, scientifically, there is a distinction in terms of the size of the bonus between the d4 chain/defended pawn and the duo pawns.
The chain/defended pawn on the same square is due around 1/3 higher bonus than the respective duo pawn, so basically the primary concept from where connected pawns derive is the defended/chain pawn, although Mr. Kmoch might think otherwise.
(Middlegame, endgame)
d4 : (15, 10)
f4, g4: (29, 23)
So SF actually scores the duo pawns a lot higher than the chain/defended pawn.
Unsupported is a different thing - c3 is also unsupported.
Could you please remind me what was the precise bonus for phalanx pawns?
Phalanx meaning duo, right?
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Haikouichthys
- Posts: 38
- Joined: Sun Oct 06, 2013 6:16 am
Re: Chain, duo, connected
Duo, yeah. If C[r] is the base bonus for a connected pawn on rank r (before the MG bonus is multiplied by 3 / 2), a phalanx has an extra bonus on top of that of (C[r+1] - C[r])/2. So, C[r] for a rank 4 pawn is 10, C[r] for a rank 5 pawn is 57, so a rank 4 phalanx pawn has the base value 10 + (57 - 10) / 2 = 10 + 47 / 2 = 10 + 23 (integer division truncates the fractional component) = 33. The MG bonus is multiplied by 3 then divided by 2, so the MG bonus is 49. As such, the full bonus for a rank 4 phalanx pawn is (49, 33). I got to (29, 23) from that by subtracting the (20, 10) unsupported penalty.Lyudmil Tsvetkov wrote:Thanks Jarrod.Haikouichthys wrote:SF does not score d4, f4 and g4 the same way. f4 and g4 have an 'unsupported' penalty and a 'phalanx' bonus. Adding all the bonuses and penalties up, we have:Lyudmil Tsvetkov wrote:Well, a chain pawn is any pawn that is part of a chain, but here we will mainly refer to chain pawns as to defended pawns, as the base/undefended pawn of the chain is much less significant and actually only due a penalty.
Duos are of course, according to Kmoch's definition, 2 own pawns next to each other on the same rank on adjacent files. A duo pawn is one of those pawns.
What concerns connected pawns, here I will refer mainly to SF's definition of connected pawns, however imprecise it might be, that considers both chain/defended pawns and duos as connected pawns, giving the same bonus.
[d]6k1/8/8/8/3P1PP1/2P5/8/6K1 w - - 0 1
above, d4 is a chain/defended pawn
f4 and g4 are a duo as a tandem, and a duo pawn each of them
SF scores both d4, and f4 and g4 in exactly the same way, as connected pawns. In practice, this works fine, as when the pawns move forward, they support each other by alternating more and less advanced duos and chain pawns, so a continuity is ensured.
However, scientifically, there is a distinction in terms of the size of the bonus between the d4 chain/defended pawn and the duo pawns.
The chain/defended pawn on the same square is due around 1/3 higher bonus than the respective duo pawn, so basically the primary concept from where connected pawns derive is the defended/chain pawn, although Mr. Kmoch might think otherwise.
(Middlegame, endgame)
d4 : (15, 10)
f4, g4: (29, 23)
So SF actually scores the duo pawns a lot higher than the chain/defended pawn.
Unsupported is a different thing - c3 is also unsupported.
Could you please remind me what was the precise bonus for phalanx pawns?
Phalanx meaning duo, right?
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Lyudmil Tsvetkov
- Posts: 6052
- Joined: Tue Jun 12, 2012 12:41 pm
Re: Chain, duo, connected
So, not to calculate extensively, could you please tell me what fraction of the main connected bonus the phalanx bonus represents?Haikouichthys wrote:Duo, yeah. If C[r] is the base bonus for a connected pawn on rank r (before the MG bonus is multiplied by 3 / 2), a phalanx has an extra bonus on top of that of (C[r+1] - C[r])/2. So, C[r] for a rank 4 pawn is 10, C[r] for a rank 5 pawn is 57, so a rank 4 phalanx pawn has the base value 10 + (57 - 10) / 2 = 10 + 47 / 2 = 10 + 23 (integer division truncates the fractional component) = 33. The MG bonus is multiplied by 3 then divided by 2, so the MG bonus is 49. As such, the full bonus for a rank 4 phalanx pawn is (49, 33). I got to (29, 23) from that by subtracting the (20, 10) unsupported penalty.Lyudmil Tsvetkov wrote:Thanks Jarrod.Haikouichthys wrote:SF does not score d4, f4 and g4 the same way. f4 and g4 have an 'unsupported' penalty and a 'phalanx' bonus. Adding all the bonuses and penalties up, we have:Lyudmil Tsvetkov wrote:Well, a chain pawn is any pawn that is part of a chain, but here we will mainly refer to chain pawns as to defended pawns, as the base/undefended pawn of the chain is much less significant and actually only due a penalty.
Duos are of course, according to Kmoch's definition, 2 own pawns next to each other on the same rank on adjacent files. A duo pawn is one of those pawns.
What concerns connected pawns, here I will refer mainly to SF's definition of connected pawns, however imprecise it might be, that considers both chain/defended pawns and duos as connected pawns, giving the same bonus.
[d]6k1/8/8/8/3P1PP1/2P5/8/6K1 w - - 0 1
above, d4 is a chain/defended pawn
f4 and g4 are a duo as a tandem, and a duo pawn each of them
SF scores both d4, and f4 and g4 in exactly the same way, as connected pawns. In practice, this works fine, as when the pawns move forward, they support each other by alternating more and less advanced duos and chain pawns, so a continuity is ensured.
However, scientifically, there is a distinction in terms of the size of the bonus between the d4 chain/defended pawn and the duo pawns.
The chain/defended pawn on the same square is due around 1/3 higher bonus than the respective duo pawn, so basically the primary concept from where connected pawns derive is the defended/chain pawn, although Mr. Kmoch might think otherwise.
(Middlegame, endgame)
d4 : (15, 10)
f4, g4: (29, 23)
So SF actually scores the duo pawns a lot higher than the chain/defended pawn.
Unsupported is a different thing - c3 is also unsupported.
Could you please remind me what was the precise bonus for phalanx pawns?
Phalanx meaning duo, right?
-
Haikouichthys
- Posts: 38
- Joined: Sun Oct 06, 2013 6:16 am
Re: Chain, duo, connected
It varies. The approximate value of a phalanx as a percentage of the value of a standard connected pawn (so 100% would mean the phalanx bonus was 0):Lyudmil Tsvetkov wrote:So, not to calculate extensively, could you please tell me what fraction of the main connected bonus the phalanx bonus represents?Haikouichthys wrote:Duo, yeah. If C[r] is the base bonus for a connected pawn on rank r (before the MG bonus is multiplied by 3 / 2), a phalanx has an extra bonus on top of that of (C[r+1] - C[r])/2. So, C[r] for a rank 4 pawn is 10, C[r] for a rank 5 pawn is 57, so a rank 4 phalanx pawn has the base value 10 + (57 - 10) / 2 = 10 + 47 / 2 = 10 + 23 (integer division truncates the fractional component) = 33. The MG bonus is multiplied by 3 then divided by 2, so the MG bonus is 49. As such, the full bonus for a rank 4 phalanx pawn is (49, 33). I got to (29, 23) from that by subtracting the (20, 10) unsupported penalty.Lyudmil Tsvetkov wrote:Thanks Jarrod.Haikouichthys wrote:SF does not score d4, f4 and g4 the same way. f4 and g4 have an 'unsupported' penalty and a 'phalanx' bonus. Adding all the bonuses and penalties up, we have:Lyudmil Tsvetkov wrote:Well, a chain pawn is any pawn that is part of a chain, but here we will mainly refer to chain pawns as to defended pawns, as the base/undefended pawn of the chain is much less significant and actually only due a penalty.
Duos are of course, according to Kmoch's definition, 2 own pawns next to each other on the same rank on adjacent files. A duo pawn is one of those pawns.
What concerns connected pawns, here I will refer mainly to SF's definition of connected pawns, however imprecise it might be, that considers both chain/defended pawns and duos as connected pawns, giving the same bonus.
[d]6k1/8/8/8/3P1PP1/2P5/8/6K1 w - - 0 1
above, d4 is a chain/defended pawn
f4 and g4 are a duo as a tandem, and a duo pawn each of them
SF scores both d4, and f4 and g4 in exactly the same way, as connected pawns. In practice, this works fine, as when the pawns move forward, they support each other by alternating more and less advanced duos and chain pawns, so a continuity is ensured.
However, scientifically, there is a distinction in terms of the size of the bonus between the d4 chain/defended pawn and the duo pawns.
The chain/defended pawn on the same square is due around 1/3 higher bonus than the respective duo pawn, so basically the primary concept from where connected pawns derive is the defended/chain pawn, although Mr. Kmoch might think otherwise.
(Middlegame, endgame)
d4 : (15, 10)
f4, g4: (29, 23)
So SF actually scores the duo pawns a lot higher than the chain/defended pawn.
Unsupported is a different thing - c3 is also unsupported.
Could you please remind me what was the precise bonus for phalanx pawns?
Phalanx meaning duo, right?
Rank 2: 167%.
Rank 3: 87%.
Rank 4: 330%.
Rank 5: 116%.
Rank 6: 140%.
Rank 7: 145%.