How far does the pawn on -4, -5 have to go to promote?
Chess on an Infinite Plane (clbamonica vs. greypenguin)

All pawns promote at the blue line in the opponent's territory. That's the 8th rank for white pawns, and the 1st rank for the black pawns. So the hawk's pawns such as on (-4,-6) have a long distance to promote.

I corrected ...H(12,8) to ...H(8,12). The easiest way to remember is (file, rank) is the same as (x,y) coordinates used in math.
1.H(8,-3)...H(8,12)
2.Q(0,-3)...C(8,6)

Just in case you are wondering, nav won

I think there's been at least one game where the players were moving really far just to be crazy, but there was no strategy to win. But when Naviary plays for real he is formidable. In fact he is the current Champion based on a tournament and other games. Right now captain "Tug" is playing Nav. We'll see if Nav keeps the crown.
This is a game of "Chess on an Infinite Plane" between clbamonica (White) vs. greypenguin (Black).
Rules:
The Pieces:
Black and White each have the following pieces (quantity and name):
1 king
1 queen
2 chancellors
2 rooks
2 bishops
2 knights
2 guards
2 hawks
24 pawns
All pieces move as in classical chess, with the "extra" three pieces moving as follows:
Chancellor (C) - Moves and captures as rook + knight.
Hawk (H) - Leaps exactly 2 or 3 squares in any orthogonal or diagonal direction. The leaping move means it can jump over other pieces.
Guard (G) - Moves and captures the same as a king but is not affected by check.
Pawns play the same and promote at the same rank as in classical chess. White pawns promote at rank 8, and black pawns promote at rank 1. Pawns can promote to chancellor, hawk, or guard in addition to queen, rook, bishop, or knight. Pawns may capture and be captured en passant with the same rules as in classical chess.
Board Setup:

Orange brackets identify the four "classical" corner squares (1,1), (1,8), (8,1), and (8,8).
There is no castling.
There is no fifty-move rule. Draws can only occur from stalemate, threefold repetition, agreement, or a proven case of insufficient material to force checkmate.
All other rules are the same as in classical chess.
Move Notation:
Numeric coordinates are used to identify piece locations as (file#, rank#). Parenthesis are used around each coordinate. Three examples of a move notation:
1) A rook moving from (8,4) to (1,4):
R(8,4)-(1,4) or R(1,4)
2) A rook moving from (1,4) and capturing a piece on (0,4):
R(1,4)x(0,4) or Rx(0,4)
3) A pawn advancing from (-1,7) to (-1,6):
(-1,7)-(-1,6) or (-1,6)
clbamonica plays White. Good luck to both players!