User fubini - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T01:55:29Z http://mathoverflow.net/feeds/user/9599 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40337/ingenuity-in-mathematics/40374#40374 Answer by Fubini for Ingenuity in mathematics Fubini 2010-09-28T19:34:45Z 2010-09-28T19:34:45Z <p>The problem:</p> <p>A rectangle R can be tiled with smaller rectangles such that</p> <ul> <li>The sides of the smaller rectanges are parallel the sides of R.</li> <li>At least one side of each of the smaller rectangle is integral.</li> </ul> <p>Show that at least one side of R is integral.</p> <p>The proof:</p> <p>Consider $\iint_{R} e^{2 \pi i (x+y)} dxdy$</p> <p>This is zero, by adding up along each small rectangle. The result follows.</p>