Question: Threefold Repetition by Geometric Transposition

I expect this question will have been asked previously. FYI: the term transposition in the title refers to board geometry and has nothing to do with move order.

Imagine the following position - FEN: knRK4/8/8/8/8/8/8/8 b

The players continue to move the pieces around the board until a similar position occurs FEN: 8/8/8/8/8/8/8/knRK4 b

After several more moves the following position is reached FEN: k7/n7/R7/K7/8/8/8/8 b and black claims a draw by threefold repetition. The player with white pieces disagees and the arbitor is called. What decision should the arbitor make?

These are not the same positions since the pieces are on different squares.  Therefore, a draw for threefold repetition cannot be claimed (unless one of those positions occurs three times).

FIDE Laws of chess (Link):

5.2 d): "The game may be drawn if any identical position is about to appear or has appeared on the chessboard at least three times. (See Article 9.2)"

From 9.2: Positions... are considered the same, if the same player has the move, pieces of the same kind and colour occupy the same squares, and the possible moves of all the pieces of both players are the same."

@Alramech If a strict interpretation of 'pieces having to be on same squares' is applied to pawnless endgames, then geometrically equivalent tranpositions are erroneously being ignored in my opinion. I believe that, with pawnless positions, the threefold repetition rule ought to apply to exactly the same set of circumstances recurring regardless of board orientation. I still suspect this to be the case, even though the wording of the rules (above) does not appear to reflect this. If I'm wrong, then the rules themselves ought be called into question. There are 8 transpositions of the original position (knRK etc...) and exactly the same move options are available from each transposition. If you were to replace the black knight with a black bishop (kbRK etc...), then there would be 4 transpositions because a bishop can only occupy one colour.

I do believe the rules are literal in the term "same squares" and are universally interpreted that way.  I do see your point: from a purist perspective, pawnless positions can be argued to be the "same".  

From a practical point of view, however, I think the rules are better without the geometric transposition.  If "geometric threefold repetition" were introduced I would imagine at least the following problems:

• Lots of additional room for error when claiming.
• Extra confusion for players of all levels.
• Additional opportunities for a winning side (or trying-to-win side) to repeat (e.g. one player is suffering or losing and moves from the king from the a1 corner to the h1 corner; the other player accidentally makes an additional "geometric repetition" as corrals the king in the new (mirrored) corner.

Yeah, this makes sense from a practical point of view in game adjudication: as long as it doesn't polute endgame study content, I think I can live with that.

This is quite interesting, I did think of geometrically identical positions before but had not associated it with threefold repetition. Castling rights might, however, still be a challenge to consider if a king and a rook remains present on the board.

From a practical standpoint there is little or no need to consider geometric transpositions (reflections or rotations) because they are likely to first trigger the 50-move rule.  If there are few enough pieces to have the transpositions occur within the 50-move rule then the player is probably clueless enough to trigger the 50-move rule anyway.

Indeed, although the pieces may be close to the centre and able to quickly move between quadrants. I would still be tempted to make a draw claim, especially if I was in time trouble.

@8

 "I would also want FIDE to explain their undocumented seventeenfold repetition rule."
++ There is no 17 fold repetition rule, only 3-fold and 5-fold.

9.2     The game is drawn, upon a correct claim by a player having the move, when the same position for at least the third time (not necessarily by a repetition of moves):

9.2.1    is about to appear, if he/she first indicates his/her move, which cannot be changed, by writing it on the paper scoresheet or entering it on the electronic scoresheet and declares to the arbiter his/her intention to make this move, or

9.2.2    has just appeared, and the player claiming the draw has the move.

9.2.3    Positions are considered the same if and only if the same player has the move, pieces of the same kind and colour occupy the same squares and the possible moves of all the pieces of both players are the same. Thus positions are not the same if:

9.2.3.1    at the start of the sequence a pawn could have been captured en passant.

9.2.3.2    a king had castling rights with a rook that has not been moved, but forfeited these after moving. The castling rights are lost only after the king or rook is moved.

9.6     If one or both of the following occur(s) then the game is drawn:

9.6.1    the same position has appeared, as in 9.2.2 at least five times.

https://handbook.fide.com/chapter/E012023 

@jetoba After some reflection, I realise that it is possible to transpose some symetrical positions by a single king move for each side: imagine four black rooks occupying all corner squares of the board and two kings waltzing around the extended centre.

I deleted that comment, but I shall explain. The number seventeen (repetitions) is a direct consequence of the same situation occuring twice in each of the eight board orientations (occuring with flip, mirror and rotate) before finally satisfying the FIDE threefold repetition rule. If one player cannot win after two attempts from the same position, then why allow play to continue from the same position again?

I will grant that.  However, if somebody is doing a symmetrical repetition in such a position then they are also likely to do a normal repetition or 50-move draw.

Well this is a thing in theory at least: any engine worth its salt will stop analysing at the first repeat and recognize all equivalent inversions.

They are, but the players won't be aware of symmetrical transposition which may repeat quicker than standard repeats like : 1. wA bB 2. w-A b-B which already permits a draw claim if wA plus bB cause a symmetrical transposition (A and B are moves,-A and -B are the same moves but inverted).

I think repeats by symmetrical transpositions are a nice challenge for puzzle composers but confusing to game players! Let's give it to the pro's as a an optional tool..