Equivalent positions

[icecream17]

Naive equivalence

If these are the same, two move lists are the same position:

• Board
• Side to move
• En passant
• Castling
• Fifty move counter
  • Implemented as unreachable position count
• Set of positions since last irreversible move (because of repetition)

Yet there is more equivalence than that

[icecream17]

Board

A 2d square has 8 symmetries:

• nothing
• horizontal
• vertical
• diagonal 1
• diagonal 2
• 90 degree rotation
• 180 degree rotation
• -90 degree rotation

i.e. these transformations do nothing as far as the chessboard are concerned

Why don't combinations of these symmetries male new symmetries?

Imagine placing an arrow at any corner of the chessboard, going from the corner to its vertical or horizontal neighbor. Since there are 4 corners, and 2 neighbors per corner, there are 8 possible arrow orientations. And since each chessboard square is relative to that arrow (and this relativity can be represented with a triangle where the arrow is one side ...), the 64 squares only have 8 constantly relative orientations, so the whole chessboard has 8 symmetries. QED. Kinda.

First of all I'm not sure vertical symmetry works:

If black were to move at the start of the game, while all of black's pieces are vertically symmetrical, black's king is to the "left" of the queen while white's king is to the "right". Does this matter?

It's non trivial to prove that these any two positions are equivalent if you haven't proven anything, but we can start with the same ideas as symmetry: If the pieces and moves correspond two symmetries are equivalent.

So first an example of symmetry:

Side to move

Compare these two positions:

Indeed the positions are equivalent.

This shows another symmetry: the color of the pieces

En passant

Note that pawn moves 2 ≠ en passant time!

Accounted for easily