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An odd chess crime has occurred. The below position, a criminal one, was on the board when it was intercepted.

enter image description here

Is it a legal or illegal chess position? Give reasonable proof as to why. You are to judge whether it is guilty (illegal) or innocent (legal)!

Hint:

The last moves could have been, among many possible ways, 1... Bh6-f8+ 2. Ka3-a2 Ba3, meaning it's White's turn to move in the diagram.

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    $\begingroup$ Sorry, I forgot to add the reason for editing, but it was to fix grammar. $\endgroup$ – CGreen Nov 20 '19 at 2:00
  • $\begingroup$ I'm no chess pro but I'm pretty sure you can't have 9 black bishops in a game. Would that be a valid answer or am I misunderstanding something? $\endgroup$ – John Zhau Nov 20 '19 at 2:26
  • $\begingroup$ @John Zhau Yes you can have 9 Black bishops in a game. There's a little something called pawn promotion. $\endgroup$ – Rewan Demontay Nov 20 '19 at 2:35
  • $\begingroup$ It's been so long since I've played that I forgot to consider that trick... $\endgroup$ – John Zhau Nov 20 '19 at 2:37
  • $\begingroup$ Is it right to assume white start on the bottom, and black from the top? $\endgroup$ – Mukyuu Nov 20 '19 at 7:40
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To start with White had 9 pieces to offer, and black has 9 bishops (5 black and 4 white) and a pawn in f3 in the end with the starting knight pair, rook pair, and a queen.

This means black pawn in A, B column was promoted in A1 while killing 1 white piece, and the black pawn in G, H column was promoted in H1 while killing 1 white piece (before white move h3). This means we got 4 leftover pawns in C, D, E, F columns while black has promoted 4 bishops (6 bishops in-game, 3 white, 3 black) and white had 7 pieces left to offer.

3 more bishops to promote while the black need to move to A1 or H1 with 7 pieces remaining?


To go from C to A column white needs to sacrifice 2 pieces. The same story goes for F to H. This means black has 8 bishops (4 black,4 white) currently. While white only had 3 pieces left to offer. Meanwhile, the end positions show that black got 5 black bishops and a pawn at f3 in the end.

The leftover pawn was D, E of black. This means E pawn needs to be promoted in E1 for it to be black, this cost white 2 pieces for a black pawn to go to F row and back to E row. Finally, D pawn needs to go to the F column and this requires white to fork out another 2 pieces. Which white missing out a piece.

Which means:

Guilty

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After a deliberation of several months, the jury has reached a verdict, and found the defendant

guilty.

Since there are very few legal options for white's latest moves, we can deduce that two moves ago the board must have looked exactly like this:

enter image description here,

It is now white's turn, and the current position was reached by these moves:

1: c3 Bc2
2: b3 Nb2

(Anything else would require that the game was in an illegal position earlier, (missed possibility edited in after OP's comment:) or that the white piece that made the previous move was just captured, which would also add a ninth captured white piece to the tally.)

This means that all the bishops must have been promoted while b2 and c2 were unmoved.

Since all black pieces are accounted for, and all the 7 missing black pawns were promoted to bishops, the unmoved b2 pawn means that there can only have been one promotion on a1.

Taking this into account, let's count the minimum number of captures required to promote the black pawns:

  • the black f-pawn is still on board
  • the black a- and b-pawns needed two captures altogether (one capture for the b-pawn to move to the a file, and another capture for one of the pawns to promote on b1)
  • the black c-pawn needed only one capture
  • because both c2 and f2 were unmoved at the time, the black d and e-pawns needed to capture three pieces in total: either white's e-pawn moved before the white d-pawn, or vice versa. In either case, the corresponding black pawn needed to capture twice.
  • Similarly, because of the unmoved f2, promoting the black g and h pawns required three captures.

Adding these up, we don't even need to check the colours of the bishops' squares to reach the verdict: black's promotions can only have happened if at least 9 white pieces were captured, but only 8 white pieces are missing.

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  • $\begingroup$ The thing is that defense has a solid objection to what the prosecutor said happened last. The judge has declared a mistrial (i.e you fell for a trap that I put in my puzzle). $\endgroup$ – Rewan Demontay Feb 24 at 18:27
  • $\begingroup$ Ah, looks like the missed possibility opens up new complications. I'll have to revisit this later. $\endgroup$ – Bass Feb 24 at 18:56

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