What's the rarest kind of checkmate?

I delivered a king discovered checkmate to a computer

fool's mate

Doubly disambiguated knight capture checkmate

Bishop and knight checkmate...where both pieces were the result of underpromotion?

Castle mate

oh i have that

this took me way to long to understand.

Has it ever happened? If it's happened more than once (or at all), it's not the rarest.

Instead Of Taking A Pawn As A Checkmate Take A Queen As A Checkmate And Your 3 3 3 Bishops Has The Be The Farthest Square Defend The E5 Square.

1 In 684 Billion. Because Take The Queen. 3 Bishops At The Farthest Square. Defending The E5 Square. Not 1 In 342 Billion.

Can You Get Rarer Than 1 In 684 Billion?

Yes. If a checkmate has been played it's not the rarest. Zero is rarer than one.

As someone else said, we have to define rare first.

Castle or En Passant

double disambiguated bishop chekmate, that's the rarest

quadratic checkmate of quadruple disambiguated knight checkmate of pq

its fools` mate (both are fools)

Rumor has it that it's happened at least once. So it can't be the rarest.