This game is a combination of chess and checkers. It was originally published in a chess fanzine [which one?]. It is played on a normal checkerboard. The setup is identical to checkers, except for the rear 4 "checkers". On the back row, in place of the four checkermen each player has two checker kings (moving and capturing as in normal American checkers), one bishop and one cook (Golomb's name for a (3,1)-leaper, more commonly known as a Camel).
When a checker man gets to the back row, it can be promoted into a bishop, cook or checker king. The bishops move and capture precisely as in chess, viz., capturing by replacement. The cooks move and capture like chess knights, except that they move to the opposite corner of a 4x2 rectangle rather than a 3x2 rectangle. Thus both bishops and cooks are colorbound to the black squares. Unlike a checker piece, bishops and cooks are not allowed to keep capturing after the first capture.
The object is to capture all the opponent's kings; stalemating the opponent is also a win.
Sources
D. B. Pritchard, The Classified Encyclopedia of Chess Variants, ch. 19 (https://www.jsbeasley.co.uk/encyc/175183.pdf)
Martin Gardner, Mathematical Magic Show, ch.14 (http://www.logic-books.info/sites/default/files/k07-mathematical_magic_show.pdf)
https://www.chessvariants.com/crossover.dir/cheskers.html
https://en.wikipedia.org/wiki/Cheskers
https://en.grandgames.net/online/chesckers/ (site allows online play)