Influence Quantity of Pieces

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Samuel Bak - Intruder [1]

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 [2]. 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
+---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+     +---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   |     |   |   |   |   |   |   |   |   |     |   |   |   |   |   |   |   |   |
| 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
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] .

 Fibonacci spiral [5] Yupana, counting tool of the Incas
Piece covers other pieces # ÷ (7x16)
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.

 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
87.9691 115.88
5 Bishop BlackBishop
The Crazy Bishop
Venus
224.70069 583.92

(579–589)

13 Queen Queen
Terra
Earth
365.256366
61 Mars
686.971 779.96

(764–811)

89 Anubis Asteroids
127 Jupiter Jupiter
4,331.572 398.88
167 Charon
181 Saturn
10,759.220 378.09
307 Chiron Chiron
18,539.000
487 Uranus
30,799.095 369.66
499 Pholus 33,547.410
547 Neptune Neptune
60,190.000 367.49
761 Pluto
90,613.305 366.73
1307 Transpluto