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'''[[Main Page|Home]] * [[Chess]] * Anti-Diagonals'''
[[FILE:antidia.JPG|border|right|thumb|link=Bibob|Main Anti-Diagonal <ref>[[Bibob]] image by [[Gerd Isenberg]]</ref> ]]
'''Anti-Diagonals''' are the diagonals from south-east to north-west on a chess board, the main anti-diagonal is h1\a8. There are 15 anti-diagonals, with line-length from 1 to 8. An Anti-Diagonal is monochrome, all their [[squares]] are either white or black.
=Square Mapping Notes=
A [[Flipping Mirroring and Rotating#Rotationby90degreesClockwise|90 degree rotation]] of the [[Chessboard]], as well as [[Flipping Mirroring and Rotating#FlipVertically|flipping vertically]] (reversed [[ranks]]) or (exclusive) [[Flipping Mirroring and Rotating#MirrorHorizontally|mirroring horizontally]] (reversed [[files]]), change the roles of [[diagonals]] and anti-diagonals. However, we define the main diagonal on the chess board from a1/h8 and the main anti-diagonal from h1\a8. Whether the square difference of neighbored squares on a diagonal or anti-diagonal is either 7 or 9, depends on the square mapping. We further rely on [[Square Mapping Considerations#LittleEndianRankFileMapping|little-endian rank-file mapping]].
=Square Difference=
Within a 0..63 square index range and the mentioned [[Square Mapping Considerations#LittleEndianRankFileMapping|square mapping]] (a1 = 0), the difference of two neighbored squares (if any) on an anti-diagonal is '''seven'''.
=Enumeration=
If we follow an anti-diagonal from south-east (h1) to north-west (a8) step by step, we increment the rank, but decrement the file, which yields in same sum. Thus, adding [[Ranks|rank]] and [[Files|file]] indices enumerates all Anti-Diagonals.
* Square a1 (file- and rank index 0) is therefor anti-diagonal with index 0 and length 1.
* The main anti-diagonal h1\a8 with index 7 and length 8.
* Square h8 is the 15th anti-diagonal with index 14 and length 1.
* All even indices are the anti-diagonals with dark [[squares]]
'''rank + file'''
{| class="wikitable"
|-
! r/f
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
|-
! 7
! 7
| style="text-align:right;" | 8
| style="text-align:right;" | 9
| style="text-align:right;" | 10
| style="text-align:right;" | 11
| style="text-align:right;" | 12
| style="text-align:right;" | 13
| style="text-align:right;" | 14
|-
! 6
| style="text-align:right;" | 6
! 7
| style="text-align:right;" | 8
| style="text-align:right;" | 9
| style="text-align:right;" | 10
| style="text-align:right;" | 11
| style="text-align:right;" | 12
| style="text-align:right;" | 13
|-
! 5
| style="text-align:right;" | 5
| style="text-align:right;" | 6
! 7
| style="text-align:right;" | 8
| style="text-align:right;" | 9
| style="text-align:right;" | 10
| style="text-align:right;" | 11
| style="text-align:right;" | 12
|-
! 4
| style="text-align:right;" | 4
| style="text-align:right;" | 5
| style="text-align:right;" | 6
! 7
| style="text-align:right;" | 8
| style="text-align:right;" | 9
| style="text-align:right;" | 10
| style="text-align:right;" | 11
|-
! 3
| style="text-align:right;" | 3
| style="text-align:right;" | 4
| style="text-align:right;" | 5
| style="text-align:right;" | 6
! 7
| style="text-align:right;" | 8
| style="text-align:right;" | 9
| style="text-align:right;" | 10
|-
! 2
| style="text-align:right;" | 2
| style="text-align:right;" | 3
| style="text-align:right;" | 4
| style="text-align:right;" | 5
| style="text-align:right;" | 6
! 7
| style="text-align:right;" | 8
| style="text-align:right;" | 9
|-
! 1
| style="text-align:right;" | 1
| style="text-align:right;" | 2
| style="text-align:right;" | 3
| style="text-align:right;" | 4
| style="text-align:right;" | 5
| style="text-align:right;" | 6
! 7
| style="text-align:right;" | 8
|-
! 0
| style="text-align:right;" | 0
| style="text-align:right;" | 1
| style="text-align:right;" | 2
| style="text-align:right;" | 3
| style="text-align:right;" | 4
| style="text-align:right;" | 5
| style="text-align:right;" | 6
! 7
|}
=Alternative Enumeration=
Some alternative enumeration of anti-diagonals to make the main-diagonal index 0, by xoring the sum with 7 (which complements the lower three bits of the sum). This yields in a 0..15 range with 8 as gap or [https://en.wikipedia.org/wiki/Nexus Nexus] in the center of the range:
'''(rank + file) ^ 7'''
{| class="wikitable"
|-
! r/f
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
|-
! 7
! 0
| style="text-align:right;" | 15
| style="text-align:right;" | 14
| style="text-align:right;" | 13
| style="text-align:right;" | 12
| style="text-align:right;" | 11
| style="text-align:right;" | 10
| style="text-align:right;" | 9
|-
! 6
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | 15
| style="text-align:right;" | 14
| style="text-align:right;" | 13
| style="text-align:right;" | 12
| style="text-align:right;" | 11
| style="text-align:right;" | 10
|-
! 5
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | 15
| style="text-align:right;" | 14
| style="text-align:right;" | 13
| style="text-align:right;" | 12
| style="text-align:right;" | 11
|-
! 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | 15
| style="text-align:right;" | 14
| style="text-align:right;" | 13
| style="text-align:right;" | 12
|-
! 3
| style="text-align:right;" | 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | 15
| style="text-align:right;" | 14
| style="text-align:right;" | 13
|-
! 2
| style="text-align:right;" | 5
| style="text-align:right;" | 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | 15
| style="text-align:right;" | 14
|-
! 1
| style="text-align:right;" | 6
| style="text-align:right;" | 5
| style="text-align:right;" | 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | 15
|-
! 0
| style="text-align:right;" | 7
| style="text-align:right;" | 6
| style="text-align:right;" | 5
| style="text-align:right;" | 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
|}
=See also=
* [[Diagonals]]
* [[Ranks]]
* [[Files]]
* [[Squares]]
* [[Intersection Squares]]
=References=
<references>
'''[[Chess|Up one Level]]'''
[[FILE:antidia.JPG|border|right|thumb|link=Bibob|Main Anti-Diagonal <ref>[[Bibob]] image by [[Gerd Isenberg]]</ref> ]]
'''Anti-Diagonals''' are the diagonals from south-east to north-west on a chess board, the main anti-diagonal is h1\a8. There are 15 anti-diagonals, with line-length from 1 to 8. An Anti-Diagonal is monochrome, all their [[squares]] are either white or black.
=Square Mapping Notes=
A [[Flipping Mirroring and Rotating#Rotationby90degreesClockwise|90 degree rotation]] of the [[Chessboard]], as well as [[Flipping Mirroring and Rotating#FlipVertically|flipping vertically]] (reversed [[ranks]]) or (exclusive) [[Flipping Mirroring and Rotating#MirrorHorizontally|mirroring horizontally]] (reversed [[files]]), change the roles of [[diagonals]] and anti-diagonals. However, we define the main diagonal on the chess board from a1/h8 and the main anti-diagonal from h1\a8. Whether the square difference of neighbored squares on a diagonal or anti-diagonal is either 7 or 9, depends on the square mapping. We further rely on [[Square Mapping Considerations#LittleEndianRankFileMapping|little-endian rank-file mapping]].
=Square Difference=
Within a 0..63 square index range and the mentioned [[Square Mapping Considerations#LittleEndianRankFileMapping|square mapping]] (a1 = 0), the difference of two neighbored squares (if any) on an anti-diagonal is '''seven'''.
=Enumeration=
If we follow an anti-diagonal from south-east (h1) to north-west (a8) step by step, we increment the rank, but decrement the file, which yields in same sum. Thus, adding [[Ranks|rank]] and [[Files|file]] indices enumerates all Anti-Diagonals.
* Square a1 (file- and rank index 0) is therefor anti-diagonal with index 0 and length 1.
* The main anti-diagonal h1\a8 with index 7 and length 8.
* Square h8 is the 15th anti-diagonal with index 14 and length 1.
* All even indices are the anti-diagonals with dark [[squares]]
'''rank + file'''
{| class="wikitable"
|-
! r/f
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
|-
! 7
! 7
| style="text-align:right;" | 8
| style="text-align:right;" | 9
| style="text-align:right;" | 10
| style="text-align:right;" | 11
| style="text-align:right;" | 12
| style="text-align:right;" | 13
| style="text-align:right;" | 14
|-
! 6
| style="text-align:right;" | 6
! 7
| style="text-align:right;" | 8
| style="text-align:right;" | 9
| style="text-align:right;" | 10
| style="text-align:right;" | 11
| style="text-align:right;" | 12
| style="text-align:right;" | 13
|-
! 5
| style="text-align:right;" | 5
| style="text-align:right;" | 6
! 7
| style="text-align:right;" | 8
| style="text-align:right;" | 9
| style="text-align:right;" | 10
| style="text-align:right;" | 11
| style="text-align:right;" | 12
|-
! 4
| style="text-align:right;" | 4
| style="text-align:right;" | 5
| style="text-align:right;" | 6
! 7
| style="text-align:right;" | 8
| style="text-align:right;" | 9
| style="text-align:right;" | 10
| style="text-align:right;" | 11
|-
! 3
| style="text-align:right;" | 3
| style="text-align:right;" | 4
| style="text-align:right;" | 5
| style="text-align:right;" | 6
! 7
| style="text-align:right;" | 8
| style="text-align:right;" | 9
| style="text-align:right;" | 10
|-
! 2
| style="text-align:right;" | 2
| style="text-align:right;" | 3
| style="text-align:right;" | 4
| style="text-align:right;" | 5
| style="text-align:right;" | 6
! 7
| style="text-align:right;" | 8
| style="text-align:right;" | 9
|-
! 1
| style="text-align:right;" | 1
| style="text-align:right;" | 2
| style="text-align:right;" | 3
| style="text-align:right;" | 4
| style="text-align:right;" | 5
| style="text-align:right;" | 6
! 7
| style="text-align:right;" | 8
|-
! 0
| style="text-align:right;" | 0
| style="text-align:right;" | 1
| style="text-align:right;" | 2
| style="text-align:right;" | 3
| style="text-align:right;" | 4
| style="text-align:right;" | 5
| style="text-align:right;" | 6
! 7
|}
=Alternative Enumeration=
Some alternative enumeration of anti-diagonals to make the main-diagonal index 0, by xoring the sum with 7 (which complements the lower three bits of the sum). This yields in a 0..15 range with 8 as gap or [https://en.wikipedia.org/wiki/Nexus Nexus] in the center of the range:
'''(rank + file) ^ 7'''
{| class="wikitable"
|-
! r/f
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
|-
! 7
! 0
| style="text-align:right;" | 15
| style="text-align:right;" | 14
| style="text-align:right;" | 13
| style="text-align:right;" | 12
| style="text-align:right;" | 11
| style="text-align:right;" | 10
| style="text-align:right;" | 9
|-
! 6
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | 15
| style="text-align:right;" | 14
| style="text-align:right;" | 13
| style="text-align:right;" | 12
| style="text-align:right;" | 11
| style="text-align:right;" | 10
|-
! 5
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | 15
| style="text-align:right;" | 14
| style="text-align:right;" | 13
| style="text-align:right;" | 12
| style="text-align:right;" | 11
|-
! 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | 15
| style="text-align:right;" | 14
| style="text-align:right;" | 13
| style="text-align:right;" | 12
|-
! 3
| style="text-align:right;" | 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | 15
| style="text-align:right;" | 14
| style="text-align:right;" | 13
|-
! 2
| style="text-align:right;" | 5
| style="text-align:right;" | 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | 15
| style="text-align:right;" | 14
|-
! 1
| style="text-align:right;" | 6
| style="text-align:right;" | 5
| style="text-align:right;" | 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | 15
|-
! 0
| style="text-align:right;" | 7
| style="text-align:right;" | 6
| style="text-align:right;" | 5
| style="text-align:right;" | 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
|}
=See also=
* [[Diagonals]]
* [[Ranks]]
* [[Files]]
* [[Squares]]
* [[Intersection Squares]]
=References=
<references>
'''[[Chess|Up one Level]]'''
