Using Column Numbers To Annotate Queen Positions
On The N-Queens-On-N x N-Chessboard Puzzle

In order to describe the positions of Queens as they appear in solutions to the "N Queens on an N x N chessboard" puzzle, I propose a simple notation schema based on an index or a 'known next coordinate' of each successive Queen.

For example, the very first solution to the N Queens on an 8 x 8 chessboard could be shown simply as:
1 5 8 6 3 7 2 4    (or optionally with comma separators - 1,5,8,6,3,7,2,4 )
This simple series of numbers contains all we need to describe the Queens on the board:
Starting at the bottom left of the board
Place the first Queen in column 1 (or File 1) of row 1 (or Rank 1), then
Place the second Queen in column 5 of row 2, then
Place the third Queen in column 8 of row 3, then
Place the third Queen in column 6 of row 4 ...
In the context of this puzzle we can assume we're describing the positions of successive Queens on the board, and therefore the position of each digit in that short sequence can imply one of the two coordinates (the Y-coordinate, or the Rank) which are needed to identify any particular Queen.
Here's a familiar 8 x 8 chessboard with the Queens placed as in the first solution to the puzzle.
Eighth Queen is in column 4
Seventh Queen is in column 2
Sixth Queen is in column 7

Fifth Queen is in column 3

Fourth Queen is in column 6

Third Queen is in column 8

Second Queen is in column 5
First Queen is in column 1
4 This is Rank 8 (or Row 8) of this 8 x 8  board
1 This is Rank 1 (or Row 1) of the board
The positions of the non-clashing Queens above can be represented
unambiguously by the simple sequence 1 5 8 6 3 7 2 4

Why Not Use An Existing Chess Notation Scheme?

My arguments for this simplified method of annotation are as follows:

Argument 1. One existing notation scheme is called Algebraic chess notation and it uses a combination of a letter and a digit to describe each square.  However this would soon be useless for our puzzle because the sizes of 'boards' for which we are now seeking solutions is already more than 26 columns by 26 rows. With only 26 letters in the alphabet we would run out of letters to describe the squares.

Argument 2. For a similar reason, the elegant ICCF numeric notation would soon become unusable because it depends on pairs of single digits as coordinates to unambiguously describe each square on the assumed normal-sized 8 x 8 board.  It then uses pairs of those double digits (totalling four digits) to describe the movement of chess pieces between squares. However, because our N x N puzzles are now looking at boards much bigger than 9 x 9, the extra digits above 9 that would be necessary for this puzzle, would soon make that scheme unusable too. (Just as an aside, the ICCF numeric notation uses a fifth digit to describe some Pawn moves, but we aren't using Pawns in this puzzle).

Argument 3. Our N-Queens-On-N x N-board puzzle involves only one of the six possible chess pieces (the Queen), and because I don't propose that my coordinate scheme should describe how Queens move from square to square, each 'column' coordinate can be represented by a simple number, coupled with the known position of that number in a series of digits.

(I acknowledge that in some notation schemes like the ICCF one, the specific chess piece is understood implicitly without needing an extra letter like B for Bishop, or R for Rook.)

Argument 4. Because the people discussing this puzzle need not necessarily be chess players (and I suggest they will likely not be chess players!), it isn't crucial for us to use a notation system familiar to them.  Therefore we have an opportunity to use this simpler index - based, or 'known next coordinate' scheme without causing unwelcome confusion.

Direction - Always Front To Back

When we attempt to solve this puzzle using computers we have the freedom to start at any board-edge, and the freedom to begin with either one Queen at a time or with eight Queens already placed somewhere on the board.  I propose that users of my notation schema take as the starting point, the 'front' edge, or row 1, and always move upwards towards the highest row, one row at a time.

(Many algorithms will also typically proceed rightwards from the left edge of the board too, but the left-right direction isn't significant for this notation schema.)
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