Common Information: “muddy children” puzzle

From paper  "Knowledge and Common Knowledge in a Distributed Environment" quoting

BARWISEJ, . Scenesand other situations.J. Philo. LXXVIII 7(1981), 369-397. :

 Classical example of this phenomenon is the muddy childrenpuzzle-a variant of the well known “wise men” or “cheating wives” puzzles.  

Imagine n children playing together. The mother of these children has toldthem that if they get dirty there will be severe consequences. So, of course,each child wants to keep clean, but each would love to see the others getdirty. Now it happens during their play that some of the children, say k ofthem, get mud on their foreheads. Each can seethe mud on others but noton his own forehead. So, of course, no one says a thing. Along comes thefather, who says,“At leastone of you hasmud on your head,” thus expressinga fact known to each of them before he spoke (if k > 1). The father then asksthe following question, over and over: “Can any of you prove you have mudon your head?” Assuming that all the children are perceptive, intelligent,truthful, and that they answer simultaneously, what will happen? 

There are two interesting proof about both the problem and a negative result without the initial information that "at least one of you has mud on your head".