Chess960 Enumbering Scheme
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The game Chess960, played with conventional chess pieces and rules, starts with a random selection of one of 960 positions for the pieces. Arrangements of the pieces are restricted so that the king is between the rooks and the bishops are on different colored squares. In order to both select a valid arrangement and to then concisely discuss which randomly selected arrangement a particular game used, the Chess960 Enumbering Scheme is used: a number between 1 and 960 indicates a valid arrangement and given an arrangement the number can be determined.
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[edit] Chess960 Enumbering Scheme
The Chess960 Enumbering Scheme can be shown in the form of a simple two tables representation. Also a direct derivation of starting arrays exists for any given number from 1 to 960. This mapping of starting arrays and numbers stems from Reinhard Scharnagl and is now used worldwide for Chess960. The enumeration has been published first in the internet and then 2004 in his (German language) book "Fischer-Random-Schach (FRC / Chess960) - Die revolutionäre Zukunft des Schachspiels (inkl. Computerschach)", ISBN 3-8334-1322-0.
[edit] Two Tables Representation
These two tables will serve for a quick mapping of an arbitrary Chess960 starting position (short: SP) at White's base row to a random number between 1 and 960 (rsp. 0 and 959). First search for the same or the nearest smaller number from the King's Table. Then determine the difference (0 to 15) to the drawn number and select that matching Bishops' positioning from the Bishop's Table. According to this first place both Bishops at the first base row, then the six pieces in the sequence of the found row of the King's Table upon the six free places left over. Finally the black pieces will be placed symmetrically to White's base row.
[edit] Example
This is the SP-518 arrangement. In the King's Table we will find No. 512 "RNQKNR". For the remainder 6 we will find "--B--B--" in the Bishop's Table at No. 6. Altogether by that for the SP-518 = 512+6 this will result in the well known white starting array "RNBQKBNR" from traditional Chess.
[edit] King's Table
| Max. | Positioning Sequence of the other Pieces | ||||||||||||||||||||||
| 0 | Q | N | N | R | K | R | 336 | N | R | K | Q | R | N | 672 | Q | R | K | N | N | R | |||
| 16 | N | Q | N | R | K | R | 352 | N | R | K | R | Q | N | 688 | R | Q | K | N | N | R | |||
| 32 | N | N | Q | R | K | R | 368 | N | R | K | R | N | Q | 704 | R | K | Q | N | N | R | |||
| 48 | N | N | R | Q | K | R | 384 | Q | R | N | N | K | R | 720 | R | K | N | Q | N | R | |||
| 64 | N | N | R | K | Q | R | 400 | R | Q | N | N | K | R | 736 | R | K | N | N | Q | R | |||
| 80 | N | N | R | K | R | Q | 416 | R | N | Q | N | K | R | 752 | R | K | N | N | R | Q | |||
| 96 | Q | N | R | N | K | R | 432 | R | N | N | Q | K | R | 768 | Q | R | K | N | R | N | |||
| 112 | N | Q | R | N | K | R | 448 | R | N | N | K | Q | R | 784 | R | Q | K | N | R | N | |||
| 128 | N | R | Q | N | K | R | 464 | R | N | N | K | R | Q | 800 | R | K | Q | N | R | N | |||
| 144 | N | R | N | Q | K | R | 480 | Q | R | N | K | N | R | 816 | R | K | N | Q | R | N | |||
| 160 | N | R | N | K | Q | R | 496 | R | Q | N | K | N | R | 832 | R | K | N | R | Q | N | |||
| 176 | N | R | N | K | R | Q | 512 | R | N | Q | K | N | R | 848 | R | K | N | R | N | Q | |||
| 192 | Q | N | R | K | N | R | 528 | R | N | K | Q | N | R | 864 | Q | R | K | R | N | N | |||
| 208 | N | Q | R | K | N | R | 544 | R | N | K | N | Q | R | 880 | R | Q | K | R | N | N | |||
| 224 | N | R | Q | K | N | R | 560 | R | N | K | N | R | Q | 896 | R | K | Q | R | N | N | |||
| 240 | N | R | K | Q | N | R | 576 | Q | R | N | K | R | N | 912 | R | K | R | Q | N | N | |||
| 256 | N | R | K | N | Q | R | 592 | R | Q | N | K | R | N | 928 | R | K | R | N | Q | N | |||
| 272 | N | R | K | N | R | Q | 608 | R | N | Q | K | R | N | 944 | R | K | R | N | N | Q | |||
| 288 | Q | N | R | K | R | N | 624 | R | N | K | Q | R | N | 960 | Q | N | N | R | K | R | |||
| 304 | N | Q | R | K | R | N | 640 | R | N | K | R | Q | N | R. Scharnagl | |||||||||
| 320 | N | R | Q | K | R | N | 656 | R | N | K | R | N | Q | ||||||||||
[edit] Bishop's Table
| Remainder | Bishops' Positioning | |||||||
| a | b | c | d | e | f | g | h | |
| 0 | B | B | - | - | - | - | - | - |
| 1 | B | - | - | B | - | - | - | - |
| 2 | B | - | - | - | - | B | - | - |
| 3 | B | - | - | - | - | - | - | B |
| 4 | - | B | B | - | - | - | - | - |
| 5 | - | - | B | B | - | - | - | - |
| 6 | - | - | B | - | - | B | - | - |
| 7 | - | - | B | - | - | - | - | B |
| 8 | - | B | - | - | B | - | - | - |
| 9 | - | - | - | B | B | - | - | - |
| 10 | - | - | - | - | B | B | - | - |
| 11 | - | - | - | - | B | - | - | B |
| 12 | - | B | - | - | - | - | B | - |
| 13 | - | - | - | B | - | - | B | - |
| 14 | - | - | - | - | - | B | B | - |
| 15 | - | - | - | - | - | - | B | B |
[edit] Direct Derivation
White's Chess960 starting array can be derived from its number as follows:
a) Divide the number by 960, determine the remainder (0 ... 959) and use that number thereafter.
b) Divide the number by 4, determine the remainder (0 ... 3) and correspondingly place a Bishop upon the matching bright square (b, d, f, h).
c) Divide the number by 4, determine the remainder (0 ... 3) and correspondingly place a Bishop upon the matching dark square (a, c, e, g).
d) Divide the number by 6, determine the remainder (0 ... 5) and correspondingly place the Queen upon the matching of the six free squares.
e) Now only one digit (0 ... 9) is left on hand; place the both Knights upon the remaining five free squares according to following scheme:
| Digit | Knights' Positioning | ||||
| 0 | N | N | - | - | - |
| 1 | N | - | N | - | - |
| 2 | N | - | - | N | - |
| 3 | N | - | - | - | N |
| 4 | - | N | N | - | - |
| 5 | - | N | - | N | - |
| 6 | - | N | - | - | N |
| 7 | - | - | N | N | - |
| 8 | - | - | N | - | N |
| 9 | - | - | - | N | N |
f) The now still remaining three free squares will be filled in the following sequence: Rook, King, Rook.
