Despite being stuck in a tiny boxing arena against both a queen and rook, the black king is a capable defender and there is no apparent mate in one without help from the white king. The closest white can do is a “series mate” in two with rook to g6, black passes, and then queen to g7 mate. Something else is needed.

Knight to d8 is a nice try, but d8 is guarded by the black rook and the king cannot block the rook. There is a spot on h7 for the white king, from which white could easily mate with the queen were white not in check. Is there any other piece that can help?

Mate is delivered by a pawn.

Black is attacking every open square on the board.

How can white deliver mate and relieve check at the same time?

This is a retrograde chess problem, wherein you must deduce information about previous moves.

Following the conventions of chess composition problems, you need to prove that black’s previous move was pawn from g7 to g5 for your solution to be legal.

Suppose that black’s previous move was pawn from g6 to g5. What was white’s move before that?

Did you consider the possibilities that white’s move was a king move, or pawn on g4 captures h5?

To show that Pg4xh5 is impossible, you need to count the number of pieces each side has and consider where else pieces may have been captured.

The king is on h4, and the winning move is h5 captures g6 en passant mate. No other square is possible because the king is in check, and no where else can white remove the check while mating.

To prove en passant is legal, we need to show that black previously moved Pg7-g5. As white is in check, the only alternative is Pg6-g5. But then what would be white’s previous move?

It could not have been a queen, rook, or knight move, as then black would have been in check on white’s turn.

The bishop and rook are enclosed and had not moved. None of the pawns could have advanced. That leaves two possibilities: a king move, or Pg4xh5.

There are four possible squares the king could have moved to h4 from, and in each of those it would have been in an impossible double check.

If the previous move was Pg4xh5, we run into a problem: since black has 14 pieces, and the white pawn on a4 must have captured a piece to get there, that accounts for all of black’s pieces; but how did the pawn get to g4 without performing any captures? It could not have started in the g file as there is still a pawn on g2.

Therefore the previous move was not Pg6-g5. Had the move been Pg7-g5, the game history would have been legal, as white’s previous move could have been either Rh7-h6 or Rg6-h6. Further rewinding is not difficult as neither king is in check so there are many possible legal sequences.

Note that the black pawns must have performed at least 3 captures (as the pawn on h2 must have captured two pieces to get around white’s pawn), which is feasible as white is missing 4 pieces. At least one of the captured pieces must have been promoted (or alternatively one of the white pieces remaining on the board), as the pawns that started on d2 and f2 could not have otherwise reached a position for black’s pawns to capture them.