## Three Kinds of Billiards Chess

A while ago, I was considering the strength of Billiards Chess pieces: they seem to be almost exactly the same strength as Cylindrical Chess pieces, so that a game in which White played Billiards Chess (sides only) while Black played Cylindrical Chess ought to be a fairly even match.

Both Cylindrical Chess and Billiards Chess are very old Chess Variants; both mentioned in Boyer.

Cylindrical Chess, if you recall, is the game where the board wraps around from side to side, so that a Rook moving to the right (Eastwards) from g1 would first cross over h1 and then arrive at a1 (and could continue from there to b1 and so on).

In Billiards Chess, the pieces bounce off the edges of the board. All of my examples will have them bouncing off the sides, but of course you can also allow them to bounce off the top and bottom.

I can't find my copy of Boyer, and so I cannot verify the actual rules of Billiards Chess as given there; however, my vague and imperfect memory is that a Bishop starting at f1 and bouncing off h3 would continue to g4, f5, and so on. I also seem to remember some illogical and incorrect rules about Knights and Pawns.

Because I lacked the official rules, I was forced to devise better rules for Billiards Chess, three sets of them in fact.

### Official Rules

Rule 0: All the rules of FIDE Chess apply except as specifically overruled by the following rules.

Rule 1: Pieces can bounce off the edge of the board. Specific details can be found in the long and rambling text that follows the official rules. The basic principle is that the angle of incidence is equal to the angle of reflection.

Note: The laws of motion must be respected. If a piece has only enough force to move one square, it cannot then bounce off the edge and thereby move another square.

Note: A capture stops your move; you can't capture and bounce back off the edge.

### Where is the Edge?

If a Knight on h1 tries to move north-northeast (aiming at the imaginary square to the right of h3), of course it bounces off the edge. But where does it land?

This is easy to figure out if we have a clear idea of where the edge is. I have distinguished three possibilities, as follows.

#### Halb-Billiards

In German, when the time is 13:30, they say that is is "halb Zwei", that is, half-two. In Halb-Billiards, the edge, and the bounce, is halfway between the edge of the board and the line that separates g1 from h1; that is, the "cushion" is right smack dab in the middle of the edge square.

If I remember correctly, this is the bounce-place of the traditional Billiards Chess. A Bishop from h1 moving northeast has effectively no bounce (it is already on the cushion, and simply moves northwest instead, to g2, f3, and so on); from g1, it goes to h2, g3, f4, e5, and so on. A Knight from g1 moving east-northeast reaches g2.

I consider this a bit repulsive because the cushion is in the middle of h1. If you have ever seen a real billiards table, you must realize that the placement of the cushion in the middle of the square means that the Rook will fall over.

#### Billiards:00

In Billiards:00, the cushion is in the most logical place, at the edge of the board.

However, you may find the results surprising. A Bishop moving northeast from h1 bounces and continues via h2, g3, f4, and so forth (It changes colors).

A Knight moving east-northeast from g1 reaches h2; moving north-northeast from h1 it reaches h3, but moving east-northeast from h1 it reaches g2.

#### Half-Past Billiards

In order to complete our collection, we have "Half-Past Billiards", where the cushion is half a square out from the edge of the board. (Of course, you could get some pretty strange effects by having the cushion be even farther out.)

You might like this because Bishops don't change color. The path from h1 is g2, h3, h5, g6, f7, e8.

A Knight on g1 cannot move east-northeast, and a Knight on h1 cannot move north-northeast. A Knight moving east-northeast from h1 arrives at h2.

### Pity the Poor Rooks

You may have noticed that Rooks do not gain anything from this board.

In fact, the Remarkable Rookies would be a very weak team in this game, and the Colorbound Clobberers would be a very strong team.

### Knights from All Four Sides

If you allow bouncing from all four sides,

in Halb-Billiards, there are no special considerations for the Knight;

in Billiards:00, a Knight moving east-southeast from g1 arrives at h1, a Knight moving east-southeast from h1 gets to g1;

In Half-Past Billiards, there are no special considerations for the Knight.

### A Useful Mapping

Billiards:00 lends itself to a mapping shortcut: you simply make a diagram that looks like this:
```  f3  g3  h3  h3  g3  f3
f2  g2  h2  h2  g2  f2
f1  g1  h1  h1  g1  f1
f1  g1  h1  h1  g1  f1
f2  g2  h2  h2  g2  f2
f3  g3  h3  h3  g3  f3
```
and now when the Knight tries to move off the board, it is easy to see exactly where it lands.

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