Influence Quantity of Pieces

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The influence quantity of pieces is defined by their number of unique moves with respect to their from- and to-square coordinates, the cardinality of the set of all possible moves, or potential, global mobility. It might be used to enumerate and encode all those moves, i.e. associating each move, per piece and in total, with a unique number for minimal perfect hashing opposed to intermittent Butterfly boards.

For instance a pawn (including promotions) has 48 (8*6) single pushes and 84 (2*7*6) captures, plus 8 possible double pushes on each file, which results in a influence quantity of 140 of either white or black pawns.The influence quantities of all pieces are divisible by four times seven (28), excluding pawn and king even by sixteen times seven (112). While obviously the number of queen quantities is the sum of rook- and bishop quantities, it is at the first glance somehow surprising that the rook quantity is the sum of bishop- and knight quantities.

=Moves per Origin= Following table gives the distinct number of moves per piece and origin square and their sums from the lower left quarter of a board (a1-d4) and in total for the whole board: Knight         King            Bishop          Rook            Queen 4 6  8  8      5  8  8  8      7  9 11 13     14 14 14 14     21 23 25 27     4  6  8  8      5  8  8  8      7  9 11 11     14 14 14 14     21 23 25 25     3  4  6  6      5  8  8  8      7  9  9  9     14 14 14 14     21 23 23 23     2  3  4  4      3  5  5  5      7  7  7  7     14 14 14 14     21 21 21 21

sum 13 19 26 26    18 29 29 29     28 34 38 40     56 56 56 56     84 90 94 96 84 = 3 * 4*7  105 = 3 *5*7   140 = 5 * 4*7   224 = 8 * 4*7   364 = 13 * 4*7   336             420            560             896            1456

Whole Board Diagrams
Whole board tables cover pawn, knight, king and sliding pieces, and their file-, rank and total sums: White pawn total                     white pawn a2                         white pawn d2 +---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+    +---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+ 13  19  19  19  19  19  19  13  140   13  15  12   9   6   3           58    6  12  15  19  15  12   9   4   92
 * 2 | 3 | 3 | 3 | 3 | 3 | 3 | 2 | 22 | 2 | 3 | 3 | 3 | 3 | 3 |   |   | 17  | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 2 | 22
 * 2 | 3 | 3 | 3 | 3 | 3 | 3 | 2 | 22 | 2 | 3 | 3 | 3 | 3 |   |   |   | 14  | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 2 | 22
 * 2 | 3 | 3 | 3 | 3 | 3 | 3 | 2 | 22 | 2 | 3 | 3 | 3 |   |   |   |   | 11  | 2 | 3 | 3 | 3 | 3 | 3 | 3 |   | 20
 * 2 | 3 | 3 | 3 | 3 | 3 | 3 | 2 | 22 | 2 | 3 | 3 |   |   |   |   |   |  8  |   | 3 | 3 | 3 | 3 | 3 |   |   | 15
 * 2 | 3 | 3 | 3 | 3 | 3 | 3 | 2 | 22 | 2 | 3 |   |   |   |   |   |   |  5  |   |   | 3 | 3 | 3 |   |   |   |  9
 * 3 | 4 | 4 | 4 | 4 | 4 | 4 | 3 | 30 | 3 |   |   |   |   |   |   |   |  3  |   |   |   | 4 |   |   |   |   |  4
 * 3 | 4 | 4 | 4 | 4 | 4 | 4 | 3 | 30 | 3 |   |   |   |   |   |   |   |  3  |   |   |   | 4 |   |   |   |   |  4

Knight                                                                     King +---+---+---+---+---+---+---+---+                                          +---+---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+                                           +---+---+---+---+---+---+---+---+ 26  38  52  52  52  52  38  26  336                                         36  58  58  58  58  58  58  36  420
 * 2 | 3 | 4 | 4 | 4 | 4 | 3 | 2 | 26                                       | 3 | 5 | 5 | 5 | 5 | 5 | 5 | 3 | 36
 * 3 | 4 | 6 | 6 | 6 | 6 | 4 | 3 | 38                                       | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 5 | 58
 * 4 | 6 | 8 | 8 | 8 | 8 | 6 | 4 | 52                                       | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 5 | 58
 * 4 | 6 | 8 | 8 | 8 | 8 | 6 | 4 | 52                                       | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 5 | 58
 * 4 | 6 | 8 | 8 | 8 | 8 | 6 | 4 | 52                                       | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 5 | 58
 * 4 | 6 | 8 | 8 | 8 | 8 | 6 | 4 | 52                                       | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 5 | 58
 * 3 | 4 | 6 | 6 | 6 | 6 | 4 | 3 | 38                                       | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 5 | 58
 * 2 | 3 | 4 | 4 | 4 | 4 | 3 | 2 | 26                                       | 3 | 5 | 5 | 5 | 5 | 5 | 5 | 3 | 36

Bishop                               Rook                                  Queen +---+---+---+---+---+---+---+---+    +---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+ 56  68  76  80  80  76  68  56  560  112 112 112 112 112 112 112 112  896  168 180 188 192 192 188 180 168 1456
 * 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 56 |14 |14 |14 |14 |14 |14 |14 |14 |112  |21 |21 |21 |21 |21 |21 |21 |21 |168
 * 7 | 9 | 9 | 9 | 9 | 9 | 9 | 7 | 68 |14 |14 |14 |14 |14 |14 |14 |14 |112  |21 |23 |23 |23 |23 |23 |23 |21 |180
 * 7 | 9 |11 |11 |11 |11 | 9 | 7 | 76 |14 |14 |14 |14 |14 |14 |14 |14 |112  |21 |23 |25 |25 |25 |25 |23 |21 |188
 * 7 | 9 |11 |13 |13 |11 | 9 | 7 | 80 |14 |14 |14 |14 |14 |14 |14 |14 |112  |21 |23 |25 |27 |27 |25 |23 |21 |192
 * 7 | 9 |11 |13 |13 |11 | 9 | 7 | 80 |14 |14 |14 |14 |14 |14 |14 |14 |112  |21 |23 |25 |27 |27 |25 |23 |21 |192
 * 7 | 9 |11 |11 |11 |11 | 9 | 7 | 76 |14 |14 |14 |14 |14 |14 |14 |14 |112  |21 |23 |25 |25 |25 |25 |23 |21 |188
 * 7 | 9 | 9 | 9 | 9 | 9 | 9 | 7 | 68 |14 |14 |14 |14 |14 |14 |14 |14 |112  |21 |23 |23 |23 |23 |23 |23 |21 |180
 * 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 56 |14 |14 |14 |14 |14 |14 |14 |14 |112  |21 |21 |21 |21 |21 |21 |21 |21 |168

Board Circles
The concentric "circles" around the center with their respective influence sums of sliding pieces: Bishop Circles                       Rook Circles                          Queen Circles +---+---+---+---+---+---+---+---+    +---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+ +   +---+---+---+---+---+---+   +     +   +---+---+---+---+---+---+   +     +   +---+---+---+---+---+---+   + +   +   +---+---+---+---+   +   +     +   +   +---+---+---+---+   +   +     +   +   +---+---+---+---+   +   + +   +   +   +---+---+   +   +   +     +   +   +   +---+---+   +   +   +     +   +   +   +---+---+   +   +   + +   +   +   +       +   +   +   +     +   +   +   +       +   +   +   +     +   +   +   +       +   +   +   + +   +   +   +---+---+   +   +   +     +   +   +   +---+---+   +   +   +     +   +   +   +---+---+   +   +   + +   +   +---+---+---+---+   +   +     +   +   +---+---+---+---+   +   +     +   +   +---+---+---+---+   +   + +   +---+---+---+---+---+---+   +     +   +---+---+---+---+---+---+   +     +   +---+---+---+---+---+---+   + +---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+ =Quantities and Point Values= The mentioned move quantities of pawns and pieces only roughly correlate with their point values since the static enumeration of all moves with distinct coordinates does not take into account the reachability of all origin-squares. Only queen and rook can reach any square in at least two moves on the otherwise empty board. Bishop quantities cover all light and dark colored from-squares. While a single bishop is bounded to one square color, its individual influence quantity is therefor divided by two. Also, pawns can only move forward and can not reach each enumerated origin square, which decreases their individual influence accordantly. King and knights can reach every square on the otherwise empty board, but may take more time with respect to distance and knight-distance.
 * 196             |     |              392              |     |              588              |
 * |         180          |   |     |   |          280          |   |     |   |          460          |   |
 * |  |      132      |   |   |     |   |   |      168      |   |   |     |   |   |      300      |   |   |
 * |  |   |   52  |   |   |   |     |   |   |   |   56  |   |   |   |     |   |   |   |  108  |   |   |   |
 * |  |   | 4*13  |   |   |   |     |   |   |   | 4*14  |   |   |   |     |   |   |   | 4*27  |   |   |   |
 * |  |    12*11      |   |   |     |   |   |    12*14      |   |   |     |   |   |    12*25      |   |   |
 * |       20* 9          |   |     |   |        20*14          |   |     |   |        20*23          |   |
 * 28* 7             |     |            28*14              |     |            28*21              |

=Divisibility by Seven= All influence quantities are divisible by four times seven, thanks to the double pushes. For a rook it is quite obvious, since each of the 64 from squares covers one rank and file each with seven squares left.

=Fibonacci Spiral= If we only consider pieces with disjoint moves (excluding pawns and king), and the queen as superset of bishop and rook, the influence quantities are even divisible by seven times sixteen, where the remaining quotients from knight to queen are Fibonacci numbers as shown by the Fibonacci spiral.

=Analogy in Astronomy= In his esoteric and pseudo scientific touched Encyclopedia of Chess-Prehistory, Peter Orantek mentions a possible connection to astronomy, related to the orbital period of Earth and Venus. The influence quantity of a queen is equivalent to about 4 years (4 x 364 days), while the influence quantity of a rook is equivalent to 224 days x 4 Venus rotations.

In his German text sample, Orantek further elaborates that queen quantities of the four center squares (four times 27) represent four earth moon rotations, while the three concentric rings around the center might related to various synodic periods of the four terrestrial planets. He associates following prime numbers with planets or objects orbiting the Sun :

=See also=
 * Butterfly Boards
 * Dumb7Fill
 * Encoding Moves
 * Move Enumeration


 * Evaluation of Pieces
 * Material
 * Mobility
 * Point Value
 * Table-driven Move Generation

=Publications=
 * H. M. Taylor (1876). On the Relative Values of the Pieces in Chess. Philosophical Magazine, Series 5, Vol. 1, pp. 221-229
 * H. S. M. Coxeter (1940). Mathematical Recreations and Essays. from the original by W. W. Rouse Ball, Macmillan
 * Jack Good (1968). A Five-Year Plan for Automatic Chess - Appendix F. The Value of the Pieces and Squares. Machine Intelligence Vol. 2
 * Dan Heisman (1990, 1999, 2010). Elements of Positional Evaluation. Russell Enterprises
 * Peter Orantek (2008). Encyclopedia of Chess-Prehistory - Programming Language Chess.

=Forum Posts=
 * Value of the pieces by Joost de Heer, rgc, February 01, 1995
 * Material/Mobility by kingliveson, OpenChess Forum, June 17, 2010

=External Links=
 * 7 from Wikipedia
 * Seven-day week from Wikipedia
 * If - Fibonacci's Number (1971), YouTube Video


 * Frank Zappa and The Mothers of Invention, Inca Roads, 1974, (A Token of His Extreme), YouTube Video
 * Frank Zappa, George Duke, Napoleon Murphy Brock, Chester Thompson, Tom Fowler, Ruth Underwood, Captain Beefheart

=References=

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