A particular combinatorial proof of Wilson's theorem I (probably re)invented a very short combinatorial proof of Wilson's theorem that I perennially teach my students.  If it occurs in the literature and someone can tell me where first (or even at all), I would like to attribute credit properly.
I actually prove $p! - p(p-1) \equiv 0 \mod p^2$. 
$p!$ counts bijective function from ${\Bbb Z}/p$ to ${\Bbb Z}/p$ and ${\Bbb Z}/p \times {\Bbb Z}/p$ acts on these by $f(x)\stackrel{(a,b)}{\rightarrow} f(x-a)+b$. Excluding functions
of the form $cx+d, c\not=0$, all orbits have size $p^2$. 
 A: Is not it just the same proof as usual (oriented, with labeled startpoint) closed broken lines, joining vertices of regular $p$-gon? Rotations correspond to shifts of the argument of the bijective function, and changing the starting point correspond to $f(x)\rightarrow f(x)+b$. 
A: I can't find my copy of Elementary number theory: A problem oriented approach at the moment but it has numerous proofs of various theorems so it would be worth a look. It is sadly out of print. It is all written in a lovely calligraphic hand. Google books actually has a searchable scan of it, but does not reveal enough to show me the proof(s) there.
later Thanks to Igor I can say that problem 14 page 64 gives the polygon argument also seen in George Andrews book along with the information that (Dickson's History V I p 75-76) says that this is due to a Danish mathematician J. Peterson (1872) and independently A. Cayley (1882).
Is it the same proof? That is theology... but let's look: ...never mind I see that others have described that. But did you know that $(m+n-1)! \equiv (-1)^nm!(n-1)! \mod m+n$?
Note: Evidently from the numbers $p! - p(p-1) \equiv\  0 \mod (p-1)p^2$. As a consequence, $(p-2)!-1 \equiv 0 \mod p.$ I suppose we know that anyway..but what are the bijections?
A: According to Dickson's History of the Theory of Numbers, this proof was first found by J. Petersen, Tidsskrift for Mathematik (3), 2, 1872, 64-5. (Petersen divides everything by 2, but the idea is the same.) Dickson gives a number of references to rediscoveries of Petersen's proof in the 19th century. Petersen was also apparently the first to discover the combinatorial proof of Fermat's theorem.
