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GroupPortrait.jpg

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 [1]. 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.
Samuel Bak - Group Portrait, 1998 [2]

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
+---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   |     |   |   |   |   |   |   |   |   |     |   |   |   |   |   |   |   |   |
| 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
|   |   |   |   |   |   |   |   |     |   |   |   |   |   |   |   |   |     |   |   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+
 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
 
Knight                                                                      King
+---+---+---+---+---+---+---+---+                                           +---+---+---+---+---+---+---+---+
| 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
+---+---+---+---+---+---+---+---+                                           +---+---+---+---+---+---+---+---+
 26  38  52  52  52  52  38  26  336                                         36  58  58  58  58  58  58  36  420
 
Bishop                                Rook                                  Queen
+---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+
| 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
+---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+
 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

Board Circles

The concentric "circles" around the center with their respective influence sums of sliding pieces:
Bishop Circles                        Rook Circles                          Queen Circles
+---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+
|              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              |
+---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+

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.

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.

Piece
#
div 4
div 7
div 28
Pawn
140
35
20
5
Knight
336
84
48
12
King
420
105
60
15
Bishops
560
140
80
20
Bishop
280
70
40
10
Rook
896
224
128
32
Queen
1456
364
208
52
Q | N
1792
448
256
64

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 [3] [4] .

spirale.jpg
Yupana_1.JPG
Fibonacci spiral [5]
Yupana, counting tool of the Incas

Piece
covers other pieces
#
div 112 (7*16)
Knight
-
336
3
Bishop
queen, king and pawn captures
560
5
Rook
queen, king and pawn pushes
896
8
possible queen move coordinates
1456
13
possible from-to move coordinates
1792
16

Analogy in Astronomy

In his esoteric and pseudo scientific touched Encyclopedia of Chess-Prehistory, Peter Orantek [6] 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.

Full_Moon_Luc_Viatour.jpg

Venus-real.jpg

The_Earth_seen_from_Apollo_17.jpg
Moon

Venus 224

Earth 364

In his German text sample [7] , 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 [8] :

Prime
number
Piece
Engine
Planet
or Objects
Symbol
Orbital period
in days
Synodic period
in days
3
Knight
Black Knight
White Knight
Mercury
Mercury_symbol.svg.png
87.9691
115.88
5
Bishop
BlackBishop
The Crazy Bishop
Venus
Venus_symbol.svg.png
224.70069
583.92
(579–589)
13
Queen
Queen
Terra
Earth
Earth_symbol.svg.png
365.256366

61


Mars
Mars_symbol.svg.png
686.971
779.96
(764–811)
89

Anubis
Asteroids



127

Jupiter
Jupiter
Jupiter_symbol.svg.png
4,331.572
398.88
167


Charon



181


Saturn
Saturn_symbol.svg.png
10,759.220
378.09
307

Chiron
Chiron
25px-Chiron_symbol.svg.png
18,539.000

487


Uranus
25px-Uranus_symbol.svg.png
30,799.095
369.66
499


Pholus

33,547.410

547

Neptune
Neptune
Neptune_symbol.svg.png
60,190.000
367.49
761


Pluto
25px-Pluto_symbol.svg.png
90,613.305
366.73
1307


Transpluto




See also


Publications


Forum Posts


External Links


References

  1. ^ Dan Heisman (1990, 1999, 2010, 2015).The Positional Elements of Chess. Russell Enterprises
  2. ^ The Game of War from Samuel Bak - represented by Pucker Gallery since 1969
  3. ^ The Fibonacci Series - The Series - Fibonacci Spiral In Action
  4. ^ The relative piece values for {N, B, R, Q} of {3, 5, 8, 15} were mentioned by H. M. Taylor (1876). On the Relative Values of the Pieces in Chess. Philosophical Magazine, Series 5, Vol. 1, pp. 221-229 as reported in H. S. M. Coxeter (1940). Mathematical Recreations and Essays. pp. 162-165, from the original by W. W. Rouse Ball, Macmillan, as reported by Jack Good (1968). A Five-Year Plan for Automatic Chess - Appendix F. The Value of the Pieces and Squares. Machine Intelligence Vol. 2
  5. ^ Peter Orantek: Encyclopedia of Chess-Prehistory
  6. ^ Peter Orantek: Encyclopedia of Chess-Prehistory - Programming Language Chess
  7. ^ Peter 0rantek: Encyclopedia of Chess-Prehistory - Decoded prehistorical books of Chess (pdf text sample)
  8. ^ Peter 0rantek: Encyclopedia of Chess-Prehistory - Decoded prehistorical books of Chess - Contents (pdf)
  9. ^ Clay animation by Bruce Bickford
  10. ^ Kelly Fisher Lowe (2006). The Words and Music of Frank Zappa. Praeger Publishers, pp. 119
    The lyrics to "Inca Roads" are absurdist in the extreme. They veer wildly from spacey - "Did a vehicle / Come from somewhere out there / Just to land in the Andes?," which both Watson and Courrier claim Zappa's satire on the popular-at-the-time book Chariots of the Gods? ...
  11. ^ Nazca Lines from Wikipedia

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