TITLE: Cutting a unit square into smaller squares
QUESTION [17 upvotes]: My algebra professor gave me this puzzle a while back.  I'm pretty sure I've found the right solution; nonetheless, I wanted to share it and see if you come up with anything really nice or unexpected.

Prove that if you take a unit square and cut it into a finite number of smaller squares (in any way you can think of), the side lengths of the smaller squares are all rational.

P.S. The first tag was my professor's hint.
[Edit] Just to be clear, every piece must actually be a square (e.g. no gluing two triangles into a square).

REPLY [6 votes]: Since your professor's hint was to use linear algebra, the intended proof is probably that of Hadwiger - employing a Hamel basis of $\:\mathbb R\:$ over $\:\mathbb Q\:$ to construct additive "area" functions in order to deduce a contradiction (as in Yuval's answer). Related results were proved by Dehn circa 1900 by different complicated methods. Although Hamel's work on Hamel bases occurred shortly later in 1905, it was not until much later in the 1950's that Hadwiger and his students noticed that Hamel bases could be employed to greatly simplify Dehn's proofs. For a nice exposition see Freiling; Rinne: Tiling a square with similar rectangles. It's worth remarking that use of the axiom of choice in the construction of a Hamel basis can easily be eliminated for applications of this type, e.g. see the discussion in section 2 of Feshchenko et. al.: Dissecting a brick into bars.