# “Short” And Sweet Math

In the following chess diagram, how many possible chess postions exist?

No looking at the solution!

Source: P1359129 & Andrew Jonathan Mestel, Retros mailing list, 2/1/2019

• I don't understand the question. What do you mean by "possible chess postions"? I wouldn't understand it even if it said "possible chess positions". – Dr Xorile Sep 24 '19 at 18:50
• I have no idea what this is asking. Is it about the number of possible next moves? As in, whose move can it be, can either side castle, are any en-passant possible? – Arnaud Mortier Sep 24 '19 at 18:52
• @ArnaudMortier Yes, it is about all of those! – Rewan Demontay Sep 24 '19 at 18:56
• I think that the lateral-thinking tag shall do @DrXorile. – Rewan Demontay Sep 24 '19 at 19:16

My first guess would be

192

because

we have the following independent options, all yielding different positions:
- White is able to castle kingside or not
- White is able to castle queenside or not
- Black is able to castle kingside or not
- Black is able to castle queenside or not
for a total of 16 possibilities

and then we can look at the last move, which yields 12 mutually exclusive options:
- it might be b2-b4 by White, in which case Black can take a4xb3 en passant;
- it might be c2-c4 by White, in which case Black can take d4xc3 en passant;
- it might be e2-e4 by White, in which case Black can take d4xe3 en passant;
- it might be f2-f4 by White, in which case Black can take g4xf3 en passant;
- it might be h2-h4 by White, in which case Black can take g4xh3 en passant;
- it might be another move by White, in which case Black is to move but no en-passant captures are possible
- it might be a7-a5 by Black, in which case White can take b5xa6 en passant;
- it might be c7-c5 by Black, in which case White can take b5xc6 en passant;
- it might be d7-d5 by Black, in which case White can take e5xd6 en passant;
- it might be f7-f5 by Black, in which case White can take e5xf6 en passant;
- it might be g7-g5 by Black, in which case White can take h5xg6 en passant;
- it might be another move by Black, in which case White is to move but no en-passant captures are possible

Of course, this assumes

those positions can be legally obtained from the starting position, but since there's no , I assume I don't have to worry about it.

I don't think

the position is possible when flipped, e.g. we're actually seeing the board from Black's position. But maybe I'm wrong and that would be the 193th to 200th position (we don't have to worry about castling rights, but en-passant captures are still possible).