Dear Sean, I would like to know if punches are legal or illegal. Please respond right away. Thanks, August G.
I want to point you to the The Official Rules of Four Square where you'll see we describe that the ball may be hit with any area between the player's wrists and fingertips, including the backs of hands. This means that the "ball may be hit with open or closed fists in the same manner as official volleyball."
So, yes, one can punch the ball. As a matter of technique, we don't recommend it. You don't have a lot of control when you punch a bouncy rubber ball and anyone who is successful in punching the ball into another square was probably more lucky than anything else. I recommend you read this blog entry about smashes, too.
And, no, you may not punch other players. That will get you out.
