# Anti-Diagonals

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Main Anti-Diagonal [1]

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 90 degree rotation of the Chessboard, as well as flipping vertically (reversed ranks) or (exclusive) 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 little-endian rank-file mapping.

# Square Difference

Within a 0..63 square index range and the mentioned 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 rank and 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

r/f 0 1 2 3 4 5 6 7
7 7 8 9 10 11 12 13 14
6 6 7 8 9 10 11 12 13
5 5 6 7 8 9 10 11 12
4 4 5 6 7 8 9 10 11
3 3 4 5 6 7 8 9 10
2 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8
0 0 1 2 3 4 5 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 Nexus in the center of the range:

(rank + file) ^ 7

r/f 0 1 2 3 4 5 6 7
7 0 15 14 13 12 11 10 9
6 1 0 15 14 13 12 11 10
5 2 1 0 15 14 13 12 11
4 3 2 1 0 15 14 13 12
3 4 3 2 1 0 15 14 13
2 5 4 3 2 1 0 15 14
1 6 5 4 3 2 1 0 15
0 7 6 5 4 3 2 1 0