Determining chess 960 starting positions with a standard die (no rethrows)

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Chess 960 is a chess game variant invented by Bobby Fischer in 1996.  

In Chess 960, the pawns are lined up on the second rank as in standard chess.   The first rank pieces are positioned randomly whilst conforming to certain caveats - the king must be placed somewhere between the rooks to allow castling; the bishops must stand on opposite coloured squares.

I've worked out a system that allows me to select a starting arrangement for the first rank pieces using a standard six sided die without the need for rethrows.  My dice procedure is different from that of Ingo Althöfer's (devised in 1998), where rethrows can occur.  Some information on how his system works can be seen here
https://www.chessvariants.org/diffsetup.dir/fischer-random-setup.html

The die I use to generate the random piece arrangement is a standard cube/hexahedron (one of the platonic solids).  Only five or six throws of the die are needed to formulate one of the 960 starting positions.

Rethrows add to the throw tally.  In my version every throw counts.  

Since the die has six faces, their numbers may be used singly or in groups of two or three to determine the positions of the first rank pieces as illustrated below...


I start with the position of the king, followed by a rook, queen, bishops and the other rook.  Knights are always placed last in the two cells that are left.  So in the first example below, numbers are marked in all the possible locations that the king can be positioned.   If a 3 is thrown the king is placed in that cell.



There are three vacant cells to the left of the king.  One of them must accommodate a rook.  Throwing a 5 or 6 will determine its starting position.



A queen is positioned in one of the remaining six vacant cells.  If a 4 is thrown the queen is placed there.



I chose the dark squared bishop next.  Throwing a 5 or 6 will result in the bishop being placed in that cell.



The light squared bishop has only two possible locations.  If a 4, 5 or 6 is thrown, the bishop is placed there.



The other rook is put into the only available cell to the right of the king (no throw indicated by a 0)



The two knights fill in the remaining two cells (no throw).



The placement order of the back row chess pieces can change slightly in the middle order.  So instead of K, R, Q, B, B, R, N, N it might be K, R, Q, B, R, B, N, N.  Here's one such example,



Does it work for all 960 starting positions?  I don't know that for sure but I don't see how it can fail either.  Below is a real life example.  I can knock these out in no time at all.



From the example above, this is how the chess pieces would be set up for a game.  Image courtesy of https://lichess.org/


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