Duo Trio Quart (Bitboards)

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**Pawn-Duo**, (Phalanx) <ref>[[http://www.chess-game-strategies.com/pawn-structure_general_page-5_phalanx-formation.html|Pawn Structure (General) | Page 5 of 5 | Phalanx Formation]]</ref> 
two adjacent [[Pawn|pawns]] of the same [[Color|color]] on the same [[Ranks|rank]] that mutually cover the other's [[Stop Square|stop square]]. A **Trio** are three horizontal friendly pawns, a **Quart** four horizontal friendly pawns.

//Working in the **bitboard centric** world to determine pawn related pattern set-wise//.

//The code snippets rely on [[General Setwise Operations#ShiftingBitboards|shifting bitboards]], specially by [[General Setwise Operations#OneStepOnly|one step only]].//

=Neighbors=
To get pawns with east or west neighbors is simple:
[[code format="cpp"]]
U64 pawnsWithEastNeighbors(U64 pawns) {
   return pawns & westOne (pawns);
}

U64 pawnsWithWestNeighbors(U64 pawns) {
   return pawnsWithEastNeighbors(pawns) << 1; // * 2
}
[[code]]
or
[[code format="cpp"]]
U64 pawnsWithWestNeighbors(U64 pawns) {
   return pawns & eastOne (pawns);
}

U64 pawnsWithEastNeighbors(U64 pawns) {
   return pawnsWithWestNeighbors(pawns) >> 1;
}
[[code]]

[[code]]
pawns               pawns with east     pawns with west     pawns with east
                    neighbors           neighbors           and west neighbors
. . . . . . . .     . . . . . . . .     . . . . . . . .     . . . . . . . .
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. . . . . . . .     . . . . . . . .     . . . . . . . .     . . . . . . . .
1 1 1 . . . . .     1 1 . . . . . .     . 1 1 . . . . .     . 1 . . . . . .
. . . . . . . .     . . . . . . . .     . . . . . . . .     . . . . . . . .
. . . . . 1 1 .     . . . . . 1 . .     . . . . . . 1 .     . . . . . . . .
. . . . . . . .     . . . . . . . .     . . . . . . . .     . . . . . . . .
[[code]]
=Neighbor Algebra=
Pawns with east or west neighbors are at least member of a duo. Pawns with east and west neighbors at least member of a trio. If two neighbors have both east and west neighbors, it is at least a quart.

An exclusive pawn duo is therefor a pawn with one neighbor, while this neighbor has no other neighbor as well.
[[code format="cpp"]]
U64 duo (U64 pawns) {
   U64 withWestNeighbors = pawnsWithWestNeighbors(pawns);
   U64 withEastNeighbors = withWestNeighbors >> 1;

   U64 withOneExclusiveNeighbor  = withWestNeighbors ^ withEastNeighbors;
   U64 withExclusiveWestNeighbor = withWestNeighbors & withOneExclusiveNeighbor;
   U64 withExclusiveEastNeighbor = withEastNeighbors & withOneExclusiveNeighbor;

   U64 duoWestOne = withEclusiveEastNeighbor & (withEclusiveWestNeighbor >> 1);
   U64 duoEastOne = duoWestOne << 1;
   return duoWestOne | duoEastOne;
}
[[code]]

[[code]]
pawns               pawns with excl.    pawns with excl.    duo
                    east neighbor       west neighbor
. . . . . . . .     . . . . . . . .     . . . . . . . .     . . . . . . . .
. . . . . . . .     . . . . . . . .     . . . . . . . .     . . . . . . . .
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1 1 1 . . . . .     1 . . . . . . .     . . 1 . . . . .     . . . . . . . .
. . . . . . . .     . . . . . . . .     . . . . . . . .     . . . . . . . .
. . . . . 1 1 .     . . . . . 1 . .     . . . . . . 1 .     . . . . . 1 1 .
. . . . . . . .     . . . . . . . .     . . . . . . . .     . . . . . . . .
[[code]]
=See also=
* [[Stop Square]]
* [[Hanging Pawns]]
* [[Pawn Islands (Bitboards)]]
* [[Phalanx]] (Engine)

=Forum Posts=
* [[http://www.talkchess.com/forum/viewtopic.php?t=52382|The phalanx concept]] by [[Lyudmil Tsvetkov]], [[CCC]],  May 21, 2014

=External Links=
* [[http://www.chess-game-strategies.com/pawn-structure_general_page-5_phalanx-formation.html|Pawn Structure (General) | Page 5 of 5 | Phalanx Formation]]
* [[https://en.wikipedia.org/wiki/Duet|Duet from Wikipedia]]
* [[https://en.wikipedia.org/wiki/Phalanx|Phalanx from Wikipedia]]
* [[https://en.wikipedia.org/wiki/Trio|Trio from Wikipedia]]
* [[http://kingaglyk.pl|Kinga Głyk Trio]] - Walking Baby, [[http://www.visioninmusica.com/|VisionInMusica]], [[https://en.wikipedia.org/wiki/Terni|Terni]], March 10, 2017, [[https://en.wikipedia.org/wiki/YouTube|YouTube]] Video
> [[Videos#KingaGlyk|Kinga Głyk]], [[http://glyk.pl/|Irek Głyk]], [[http://www.visioninmusica.com/kinga-glyk-happy-birthday/|Rafal Stepien]]
> [[media type="youtube" key="Xmq52kc_HM0"]]
* [[https://en.wikipedia.org/wiki/Quart_%28disambiguation%29|Quart (disambiguation) from Wikipedia]]
* [[Videos#PantaRhei|Panta Rhei]] plays [[https://en.wikipedia.org/wiki/B%C3%A9la_Bart%C3%B3k|Bartók's]] Quarts, 1977, [[https://en.wikipedia.org/wiki/YouTube|YouTube]] Video
> [[media type="youtube" key="xDiKq9Wf0UQ"]]

=References=
<references />
=What links here?=
[[include page="Duo Trio Quart (Bitboards)" component="backlinks" limit="20" ]]
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