Some tentative thoughts:

## Preliminaries


A checkmating move is either a discovered check, or a direct check. (Possibly both, but likely not relevant). Any proposed checkmate move must consider each possible form.

* B,N,R,P: can direct check or discovered check (unless starting in a corner, or moving along an edge in the case of R).
* K: can discovered check
* Q: can direct check

AFAIK, any possible move is a candidate for a checkmate except:

* a K move along the side of the board
* a B move from one corner to the other

There aren't many moves that really force the opposing king to be in a particular spot. Pawn moves that finish on the edge of the board (eg, a3-a4# or b3xa4#) are the only exceptions (in this case, the king must be at b5). Many moves (say Bb2c3#) are more easily reached with a discovered check (say, a rook along the b file or 2nd rank).

Assuming for the moment that White is mating, the number of half moves required can be described as:

* White: (moving mating piece into position) + (getting other pieces out of the way of mating piece) + (capturing pieces to help king) + (moving pieces into position to attack king's escape squares) + 1 (mating move)
* Black: (getting king into position) + (moves capturing pieces to help mating piece) + (moving pieces into king's escape squares)

## Possible goals

It's not immediately obvious which of these mutually incompatible goals is most important:

* forcing the king to move to an inconvenient square far away
* forcing the mating piece to do the same
* choosing a mating location that requires a lot of pieces to enclose (such as the centre of the board, rather than on an edge, or next to where other pieces start)

A good way of forcing the black king to move a long way is for the mating move to be a king move from rank 1 to 2 or similar - so the discovered attack is along a rank.


## Unpromising avenues of exploration

* Pawn promotions: it only takes 5 moves, and the goal is close to where the mated king starts, where it is already fairly trapped by the edge of the board and its own pieces.
* Queen moves: hard to see how this would ever take longer than a rook or bishop.


## Some possible candidate moves (with White mating)


### Kb2xb1

White king has to move to an inconvenient square, moving several pieces out of the way. Black king has to move all the way from 8th rank to probably d2. White rook has to get to a2 most likely. Black and white also have to set up other pieces to enclose d2. And black also has to get some other piece all the way to b1 in order to be captured.

I'm picturing a final board something like this, where the black king captures the b knight, the black knight captures the a rook, and the pawn at e4 was previously at e2, allowing the king in past the bishop.

    rnbq1b1r/pppp1ppp/4p3/8/2PBP3/6N1/QK1kBP1P/1n5R w - - 0 1

Absolute minimum moves required for:

* Black: 7 + 4 + 1 (king, knight, pawn) [maybe better with bishop instead of knight)
* White: 4 for king, 2 for queen, and who knows how many pawn and minor pieces :)

Although a very different way of achieving that same move might look like this:

    rnq3bk/ppp4p/3p2p1/5p2/P3p3/1P6/RKPPPPPP/Bn1Q1BNR w q - 0 1