This brief video shows how to play the new chess variants available on the new Chess.com, which include 3 Check, King of the Hill, and Crazy House. For now, until more people start playing, it will take a little while to get your seek matched. I recommend putting out several seeks for different time controls to get a game.
The new live chess server and the new games are available to all members (paid and free).
The goal of 3 Check is to be the first to check your opponent three times. It doesn’t matter if you’re just sacrificing pieces to do it–any check counts. Of course you have to move the pieces legally (no jumping rooks over pawns, etc.).
The goal of King of the Hill is to be the first to get your King to one of the four center squares. As with 3 Check, material and other standard chess considerations do not matter, because checkmate is not the goal.
Crazy House still involves some standard chess ideas because the goal is to give mate, but, the catch is that, any time you capture a piece, you can drop it on the board on your next turn to add it to your army. Of course, your opponent can do the same when he captures your pieces. The name Crazy House seems fitting, as these games can get pretty insane.
IM Danny Rensch has some videos showing examples of playing the games. There are a couple videos in the video archive for Diamond Members, but I believe there is also a video or two on the Chess.com YouTube channelthat anyone can access. You can always just log into the new v3 Live Chess and create some seeks to check them out for yourself.