## Introduction

In shogi most pieces have two way symmetry as opposed to eight way as in chess. As a result there are literally more than 15,000 different possible shortrange shogi pieces as oppose to just 31 chess ones. In the exposition I'll present some enumerations, classifications as well some further remarks. Along the way some insights into value might be gained, hopefully. For sake of simplicity this page will deal with pieces without accounting for promotions, on another page some info on promotions and combinatorics will appear.

## Enumerations

*First Pass*

Let's first use the rules below, distilled from the panoply of historical shogi variants^{1}:

- A piece must have left-right symmetry
- A piece must be able to go forward as well as retreat
- A piece must be approachable

- Approachable
- A piece that an opponent piece following the same rules may threaten without being threatened as well.

With these conditions let's go to the tableaux below, where each number or letter indicate a position where a piece can go or not depending on the status. The status of like number or letter indicates synchrony which both the same status.

`6 5 4 5 6
3 2 1 2 3
z y . y z
c b a b c
f e d e f`

As you can see, going forward we have six distinct positions, 1 through 6, after symmetry, so there is 2^{6}, or 64 possibilities. However we have to discount the no forward movement option, so we are left with 63. Similarly for going backward we have positions, a through f. So after applying our rules we have again 63 possibilities. The rules only constrain the symmetry of sideways movement, not their existence, so there is the full 4 possible configurations. To arrive at the full number we simply multiply the possibilities of each sub-domain, so we have 63x4x63 which is 15,876 possible pieces. Now we apply the last rule, that a piece must be approachable. The only non-approachable piece is the piece that can go to all positions. So after deducting that piece we have 15,875 pieces.

*Weight*

With 15,875 pieces, it is difficult to list every single piece, but this section and the following will layout ways to cut them into more manageable pieces. First let's define some terms that will be useful in this section and later.

Weight^{.}- Number of places a piece may move to in a move.
Offense^{.}- Number of places a piece may move forward.
Defense^{.}- Number of places a piece may move backward.
Symmetrically Reduced Tableaux^{.}- A tableaux where only a part is shown because symmetry dictate the rest.

To establish the weight of a piece we only need to look at the right half, due to symmetry. In the symmetry reduced tableaux below the number indicates the multiplier, and I have truncated it to the right half, as indicated:

`1 2 2
1 2 2
. 2 2
1 2 2
1 2 2`

As you can see, the line that goes vertically through the piece has the multiplier 1, since they are on the crease and have no dual on the other side. For places off the crease, the multiplier is 2, since there is a symmetrical pair on the other side. So for example a piece as follows, again only showing the right side, with 'x' indicating places that the piece can go, '*' where it can't and '.' indicating the piece itself.

`* x *
x * x
. x *
x * x
* x *`

Matching the xses with the numbers and adding them we get 12^{2}.

By using the weight, we slice the multitudes into 22 categories, from a count of 2 to a count of 23, which averages about 722 per category. Below is an enumeration on a category by category basis:

Weight | Number of Pieces^{3} |
---|---|

2 | 4 |

3 | 20 |

4 | 65 |

5 | 160 |

6 | 330 |

7 | 580 |

8 | 905 |

9 | 1260 |

10 | 1590 |

11 | 1824 |

12 | 1918 |

13 | 1844 |

14 | 1630 |

15 | 1320 |

16 | 975 |

17 | 660 |

18 | 400 |

19 | 220 |

20 | 106 |

21 | 44 |

22 | 16 |

23 | 4 |

*Balance*

Balance refers to the relative strength of a piece's offense to its defense, as defined below:

Balance- A piece's offense minus its defense.

By this definition and rules above, balance ranges from -9 to +9^{4}, and the corresponding number of pieces is enumerated below.

Balance | Number of Pieces^{5} |
---|---|

-9 | 8 |

-8 | 36 |

-7 | 112 |

-6 | 268 |

-5 | 528 |

-4 | 888 |

-3 | 1312 |

-2 | 1716 |

-1 | 2008 |

0 | 2123 |

1 | 2008 |

2 | 1716 |

3 | 1312 |

4 | 888 |

5 | 528 |

6 | 268 |

7 | 112 |

8 | 36 |

9 | 8 |

*Slide*

Slide is defined as follows:

Slide- Total number of places that a piece can move to the side.

Due to the demand of symmetry, there's only three possible value for slide: 0, 2, 4. The number of pieces with corresponding slide is summarized below:

Slide | Number of pieces |
---|---|

0 | 3969 |

2 | 7938 |

4 | 3968 |

### Interstitial 1

Weight, balance, and slide give some rough insights into the pieces, though sometimes very rough. However with these three dimensions, we have average of 13 pieces per combo of numbers. Enumerating pieces along these dimensions, would go a long way in teaming the chaos. Moreover, even though rough, it gives a good enough guide to come up with plausible assembles for a game with **equal armies**. And for **unequal armies**, it provides a starting point.

For example we could say: we want an assemble whose balance range from -3 to 3 and is composed of 1/2 weak pieces with weight less than 8, 1/3 composed of medium pieces with weight between 8 and 16, and 1/6 strong pieces with weight greater than 16. Which all have a slide of 2.

Another useful note is that the value of a piece when balance is not too far from zero is roughly $3^{\frac {w-2} {7}}$ where w is the weight of the piece.

Going beyond weight and balance, I will present a scheme that allows one to completely describe a piece^{6} next.

*SSR*^{7} Genotype

^{7}Genotype

A piece can be completely described by 7 bases, which I'll describe below:

Let's return to the symmetrically reduced tableaux, this time we first divide it in to seven parts, labeled ‘1’ through ‘7’

`2 4 6
1 3 5
. 7 7
1 3 5
2 4 6`

As you can see each part has 2 locations, so let us divide the tableaux in to two parts, indication the (s)uperior location and the (l)esser location:

`s s s
s s s
. l s
l l l
l l l`

Combined it looks like this:

`2s 4s 6s
1s 3s 5s
[] 7l 7s
1l 3l 5l
2l 4l 6l`

Note: [ ] represent the piece

So now all the distinguished positions can be named by number and whether it is superior or lesser. We also notice that each numbered part has four states it can be in. and label these states ‘C’, ‘T’, ‘A’, and ‘G’ respectively:

**C**a piece can't move to places in that part**T**only to the lesser position**A**only to the superior position**G**can move to both

Now if we write the states for each part in order from ‘1’ to ‘7’ then we have a complete description of the piece. This description we call the genotype^{8} of the piece.

For example:

`* x x
x * x
. x *
x * x
* x *`

Has ‘x’s at , 1s, 1l, 4s, 4l, 5s, 5l, 6s, 7l so the genotype is GCCGGAT^{9}.

Now we translate the rules to apply to the genotype.

- Since we used the symmetrically reduced tableaux, symmetry is already assumed.
- In the first six bases, at least one of the following is true:
- At least one is ‘G’
- There is a ‘T’ and an ‘A’.

- At least one of the seven bases is not ‘G’.

This system is fairly good for SSR discussed on this page but has its limitations. First it doesn't scale to certain sizes, such as pieces whose footprint is 7x7. Also left out is promotions. On the page on promotions, another scheme will be presented that handles scaling and promotion. Nonetheless for describing SSR pieces and its movement without reference to promotion, this system is more than apt.

## Toric Considerations

In Enumerations, We looked at pieces as defined by rules laid out in First Pass. As defined below, these rules are an example of a piece system definition^{10}.

Piece System Definition^{.}- A set of rules that all pieces of a group obeys.
Piece System^{.}- The rules along with the pieces that obey them.

In the sections below I will show some other examples of piece systems, both of which can be of use on a toric board. We can imagine having pieces starting out as non-retreating pieces so they don't attack each other in the opening on a toric board, and later the pieces may promote to full toric pieces. Hopefully, it will give you an idea how a piece system can color and deepen a game.

*Non-retreating Pieces*

Non-retreating pieces obeys the following rules:

- A piece must have left-right symmetry
- A piece must
**not be able**to retreat - A piece must
**be able**to move forward - A piece must be approachable

Non-retreating pieces are much easier to enumerate, they only have two parameters, weight and slide. Also because of the no-retreating rule, the symmetrically reduced tableaux can be reduced even further by chucking the retreating section, as below:

`a b c
d e f
. g h`

This we call the reduced tableaux. It has eight distinct positions, with six forward and two to the side. The forward positions gives 63 possibilities after discounting the no forward option. Side give another 4 possibilities. So total, bearing in mind the approachability rule, is 63 x 4 - 1 = 251 possibilities.

Enumeration by Weight is as below:

Weight | Number of Pieces^{11} |
---|---|

1 | 2 |

2 | 5 |

3 | 12 |

4 | 20 |

5 | 30 |

6 | 35 |

7 | 40 |

8 | 35 |

9 | 30 |

10 | 21 |

11 | 12 |

12 | 7 |

13 | 2 |

Enumeration by Slide:

Slide | Number of pieces |
---|---|

0 | 63 |

2 | 126 |

4 | 62 |

*Full Toric Pieces*

Full Toric Pieces are very much like shogi shortrange outlined in First Pass with just a slight toric twist. They obey the following:

- A piece must have left-right symmetry
- A piece must be able to go forward as well as retreat
- A piece must be approachable
- A piece must be able to tour a prime square toric board.

It can be proven that the fourth rule is equivalent to stipulating that a piece can move in two non-collinear directions. This results in the deletion of nine pieces from standard shogi shortrange. These deleted pieces has genotype XYCCCCC. where X,Y are independently C, G, T or A. This means there are 15,866^{12} full toric pieces.

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