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Here’s a simple challenge.

Wtih black to cooperate and move first, and white to checkmate, checkmate black in 6 plies (half-moves.) Only one illegal move may be made, and white makes only legal moves.

State what the illegal move is.

You must also prove that the move is illegal, i.e. retrograde analysis is nessacary to solve this puzzle.

Bonus Task (Optional, for those who would like to do this!): Construct a proof game that leads to the given position as additional proof.

Bonus Bonus Task (Super-Optional!): Find the shortest way possible to the current position! If you want to switch the colors to save half a move, you may. Rearranging the pieces is also allowed if it leads to a faster way to a similar position.

The Position:

enter image description here

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  • $\begingroup$ If black moves first, then any odd number of half-moves makes black move last as well, but you say the game should end with W checkmating B. Is this deliberate (and e.g. the intended illegality is that one player moves twice in a row) or is there some mistake here? $\endgroup$ – Gareth McCaughan Apr 20 at 0:30
  • $\begingroup$ There was a mistake! I meant to type 6, not 7! Thank you! $\endgroup$ – Rewan Demontay Apr 20 at 0:35
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Here is a solution, or more precisely a family of solutions.

1. ... O-O-O [illegal; see below]
2. Rh2 Rh8
3. any any
4. Rxh8
There's lots of freedom to choose those "any" moves; or e.g. we could have 2. Rg2 instead and have B move his R off the back rank on move 3. The main point here is that Black can't castle, for the following reason.
W's pawns need to have made at least six "leftward" captures (proof: in any position, define X to be the sum of the "column numbers" (1..8) of the white pawns or, for captured or promoted white pawns, of where they were immediately before disappearing; in the initial position X is 1+...+8=36; in this position X is at most 1+2+3+3+4+4+5+8=30, and the only way for X to decrease is via a "leftward" pawn capture).
Exactly 8 of B's pieces are no longer present on the board. If B can castle then two of those pieces -- the queen and the bishop that was initially on c8 -- must have remained on {b8,c8,d8} and therefore can't have been captured by white pawns. So all the other absent black pieces must have been captured "leftward" by white pawns.
One of those pieces is the h-pawn. A capture on the h-file can't be "leftward", so that pawn must have made a leftward capture or been promoted before its capture. It can't have made any sort of capture, though, because white still has 16 pieces. And if it promoted without capturing then white's h-pawn must have got out of the way by making a capture of its own, requiring at least seven leftward captures, which certainly isn't possible if the BK hasn't moved (because there are only six pieces available to have been captured).
So, black's king must have moved after all, letting the BQ out to be captured. And therefore black cannot castle.

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  • 2
    $\begingroup$ "You must also prove that the move is illegal." =/= "That implies that retrograde analysis is nessacary [sic]". Moving your queen like a knight would be illegal, but retrograde analysis isn't what you would use to prove that. I think this puzzle has excellent potential, but I do think that you sometimes erroneously conclude that your conditions are completely clear and lead to a unique solution. Just a friendly reminder to make sure that the answer is as logical and unique to an outside observer as it is to you. @RewanDemontay $\endgroup$ – Brandon_J Apr 20 at 1:37
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    $\begingroup$ I completely agree with what Brandon_J said about proving things. I assure you I can prove that 1. ... Kc8 is illegal without retrograde analysis :-). $\endgroup$ – Gareth McCaughan Apr 20 at 1:38
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    $\begingroup$ (No crushing of any sort intended, obviously.) $\endgroup$ – Gareth McCaughan Apr 20 at 1:38
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    $\begingroup$ Start the game. Write down the number 36. This equals the sum of the "file numbers" of all white's pawns. Every time white makes a pawn capture, add or subtract 1 depending on whether the pawn moves right or left. At all times, the number you have will equal the sum of pawns' "file numbers", with the convention that once a pawn is removed from the board you hold its "file number" constant. In the position shown, this number is 22 + whatever file number is appropriate for the one missing pawn (which must have promoted to a rook). [...continues] $\endgroup$ – Gareth McCaughan Apr 20 at 15:18
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    $\begingroup$ That file number can't possibly be >8. Therefore our number is no bigger than 30. It only decreases when a leftward (white) pawn capture is made, therefore there must have been at least 6 leftward pawn captures made by white. This "applies to retroanalysis" because it's a way of figuring out what has happened in the past: in this case, figuring out some things about pawn captures white must have made. $\endgroup$ – Gareth McCaughan Apr 20 at 15:19

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