[kgjszhlo] KNNNK

some geometrically pretty chess endgame initial positions: white king and 3 knights in corners, black king in the center of the board.  we give positions in FEN notation.

white to move, win in 20:
N6N/8/8/4k3/8/8/8/K6N w - - 0 1

same position, black to move, draw.  after Ke5-f6, the white knight on h8 is trapped then captured.
N6N/8/8/4k3/8/8/8/K6N b - - 0 1

tangent: giving black additional firepower, a pawn, can counterintuitively cause black to lose via the famous Troitzky-line KNNKP endgame.  the longest I could find (ruining the geometric symmetry) was black to move, lose in 95:
N6N/8/8/4k2p/8/8/8/K6N b - - 0 1

the other corner-and-center symmetric position:

white to move, win in 19:
N6N/8/8/8/3k4/8/8/K6N w - - 0 1

black to move, lose in 20:
N6N/8/8/8/3k4/8/8/K6N b - - 0 1

the first position is on the path of most resistance from the second position.