The term back-rank mate in chess is used to define a constraint that a king can not escape from the attack of opponents piece which results in a checkmate. Back-rank mate occurs while the king is at first or eighth rank.
The king can not move because it it blocked by its own pieces at the second at seventh rank.
Example of Back-rank mate
In the back-rank mate shown below, the black king is under attack from the white rook at d8. The king is is in eighth rank and it can not move or escape from the attack since it is blocked by the friendly pieces, i.e, pawns at seventh rank. This is called back-rank mate.
The white king at g1 can not fall into the back-rank mate since, if given a check in first rank, the king can move to f2 which is free.
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