Dissection of square into triangles

Solution 1:

The only proof I know of is the $p$-adic one due to Monsky.

EDIT: I found some confirming evidence at Math Reviews. The review of Charles H. Jepsen and Paul Monsky, Constructing equidissections for certain classes of trapezoids, Discrete Math. 308 (2008), no. 23, 5672–5681, MR2459386 (2009h:05053), written by Sherman Stein, says, in part,

The simple question, "Can a square be cut into an odd number of triangles of equal areas,'' raised some forty years ago, has generated more questions, only some of which have been answered. The answers so far have used valuations and algebraic integers.