$32$ Goldbach Variations - Papers presenting a single gem in number theory or combinatorics from different point of view

Solution 1:

Fourteen Proofs of a Result About Tiling a Rectangle collects $14$ proofs of the fact that a rectangle tiled by rectangles each of which has at least an integer side has an integer side.

Solution 2:

Elisha S. Loomis, The Pythagorean Proposition, contains $370$ proofs of the Pythagorean theorem. ERIC has a PDF of NCTM reissue of the $1940$ second edition.

Solution 3:

Robin Chapman gives many proofs of $\sum n^{-2}=\pi^2/6$.

Solution 4:

The book "The Fundamental Theorem of Algebra" by Fine and Rosenberger (link) contains detailed discussions of at least six proofs of this theorem, all rooted in different areas of mathematics. Links to other papers (not all in English) compiling various proofs of the theorem can be found at this MathOverflow question.

H. W. Kuhn gave a combinatorial proof in 1974.