There is a certain age when a kitten is at its most playful and its mind is most flexible, and the kitten loves for you to play with it, and wave around its favorite toy so it can rear up and claw at it, and sometimes you move the toy around and around near the kitten's tail and it chases and chases until it is dizzy and its little sides are heaving for breath and you worry for its well being.
If you throw the toy, the kitten will chase it down and play with it a bit but now the game is not nearly so much fun, without you. If you always end a play session by throwing the toy, the kitten will soon carry the toy back to you so that you will throw it again.
You believe that you have taught the kitten to fetch, but the kitten believes that it has taught you to throw the toy when it is brought to you. Who has the right of it?
Twenty years later, when the aged cat, feeling kittenish, brings its favorite toy to you and drops it at your feet (not having done so for years), you will understand who has taught whom.
Fetch Chess is also a two-part-move game, as in Avalanche Chess, in which first you make a normal doublemove an then the cat fetches a Pawn if it can.
Therefore, in Fetch Chess, a Pawn which has advanced from e2 to e5 in one turn can be fetched back to d4, e4, f4, c3, d3, e3, f3, g3, b2, c2, d2, e2, f2, g2, or h2, providing only that the destination square for the fetch is empty. There can also be a null fetch, leaving the Pawn at e5; cats cannot be predicted.
Both would be good games.
However, the best Cat is a third person. Fetch Chess shines when there are three persons physically present, two play each other, and the third acts as the Cat for both players. In a social setting, the third person might well be somebody who barely knows the moves of Chess. A small child would make a good Cat.
Sometimes in a chess club, you have an odd number of players, and three form a group: winner stays seated, loser gets up, kibitzer plays next. Fetch Chess should be great in this situation.
I would not have gone to the trouble of writing up Fetch Chess except that it looks to be a very promising three-person game. There are few such.
However, if you get out of check but the Cat puts you back in check, normally you should lose right away because "if you are in check and it's your opponent's turn to move, you have lost" (I think that's in Rule Zero.)
I fear this gives the Cat too much power. Thus, I add the special rule that the King may not be captured in such situations. It can still be dreadful if the Cat puts you in check -- your opponent can capture some pieces or add another check that might change it to checkmate -- but at least it is some limitation on feline power.
A promoted Pawn is no longer a Pawn, and cannot be fetched.
Perhaps this should be added because it is only fair to give credit where credit is preapproved, or however the old proverb has it.