# That Ain’t Right.... #3

Here’s a simple challenge.

With 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 necessary to solve this puzzle.

The Position:

• 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? Apr 20 '19 at 0:30

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.

• "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 Apr 20 '19 at 1:37
• 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 :-). Apr 20 '19 at 1:38
• (No crushing of any sort intended, obviously.) Apr 20 '19 at 1:38
• 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] Apr 20 '19 at 15:18
• 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. Apr 20 '19 at 15:19