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Muller: Winther is very good about making Zillions rules files for his pieces, so, if you have Zillions (it’s only $25 and excellent for prototyping variants—buy it if you haven’t done so yet), you can see what moves a given piece of Winther can do if you have any questions.

In terms of bifurcators, I assume we’re taling about a piece that:

• Moves in a straight line, either orthogonally (rook-like) or diagonally (bishop-line).
• Hits another piece, either friendly or enemy
• Changes direction, either 45 degrees or 90 degrees upon hitting the other piece
• Finishes its move

Now, given these parameters, we have a number of new interesting pieces. To keep things simple, I will only look at pieces that turn 45 degrees on hitting the other piece. So, that gives us the following pieces (# is the bifurcator to move, X is another piece, either friendly or enemy, 1 is the first part of the piece’s move, 2 is the second part of the piece’s move, and . is an empty square)

. . . . . 2 . .        . . 2 . . 2 . . .
2 . . . 2 . . .        . . 2 . . 2 X . .
. 2 X 2 . . . .        . X 2 . . 1 2 2 2
. . 1 . . . . 2        2 2 1 . 1 . . . .
. . 1 . . . 2 .        . . . # . . . . .
1 1 # 1 1 1 X .        . . 1 . 1 . . . .
. . 1 . . . 2 .        . 1 . . . 1 2 2 2
. . 1 . . . . 2        1 . . . . 2 X . .

2 . . . 2 . . .        . 2 . . . . 2 . .
. 2 . 2 . . . .        . 2 . . . . X 2 2
. . X . . . . .        2 X . . . 1 . . .
. . 1 . . . . .        . . 1 . 1 . . . .
. . 1 . . . . 2        . . . # . . . . .
1 1 # 1 1 1 X .        . . 1 . 1 . . . .
. . 1 . . . . 2        . 1 . . . 1 . . .
. . 1 . . . . .        1 . . . . . X . .

. 2 . 2 . . . .        2 . . . . . 2 . .
. . 2 . . . . .        2 . . . . . 2 2 2
. . X . . . . .        . X . . . X . . .
. . 1 . . . . .        . . 1 . 1 . . . .
. . 1 . . . . 2        . . . # . . . . .
1 1 # 1 1 X 2 .        . . 1 . 1 . . . .
. . 1 . . . . 2        . 1 . . . 1 . . .
. . 1 . . . . .        1 . . . . . X . .

Here, we see three types of these bifurcators:

• The piece bounces just before the other piece’s square (first row)
• The piece bounces in the middle of the other piece’s square (second row)
• The piece bounces just after the other piece’s square (third row)

Now given these six pieces, we can give these pieces four different powers:

• The piece can move to an empty square on the first leg of its move (the second leg is not used)
• The piece can capture on an enemy-occupied square on the first leg of its move (the second leg, again, is not used)
• The piece can move to an empty square on the second leg of its move.
• The piece can capture on an enemy-occupied square on the second leg of its move.

Pieces that can neither move or capture on the second leg of their move are nothing more than FIDE rooks and bishops, so are not interesting for our purposes. This leaves us with 12 types of powers for the pieces in question. With six types of movement for the bifurcators, this gives us 72 different types of pieces.

I’ll pull a Betza and create a notation so we can quickly describe a bifurcator. O means we start with an orthogonal move; D means we start with a diagonal move. B means we bounce just before the other piece, M means we bounce in the other piece’s square, and A means we bounce afterwords. 1m means we can end our move on the first leg, 1c means we can end our capture on the first leg, 2m means we can end our move on the second leg, and 2c means we can capture on the second leg.

OK, so where do Winther’s pieces fit in this Betza-esque scheme? Like this:

• Gladiatrix OB2m2c
• Crossrook DA1m2c
• Crossbishop OA1m2c
• Murmillo DB1m2c (also can bounce off the edge of the board)
• Secutor OB1m2c
• Provocator DB2m2c (also can bounce off the edge of the board)
• Diamachaer OB2m2c
• Sagittar DA2m2c
• Venator OA2m2c
• Laquear DB1c2m (also can bounce off the edge of the board)
• Essedar OB1c2m
• Gaul DA1c2m (not allowed to end its move in the square immediately after the second piece)
• Thraex OA1c2m (not allowed to end its move in the square immediately after the second piece)

OK, there are some other pieces that don’t fit in this scheme, but this makes a reasonable introduction to these types of pieces.