Solution to Problem 264

Correct solutions came from Rafael Stern, USA; Philippe Fondanaiche, France; Bill Webb, USA; David Stigant, USA; Al Zimmermann, USA; Brett Hendricks, USA.

Let x be the length of the longer side of the rectangle; see the figures on the left.  If the initial rectangle is similar to the one left after clipping off the red square, then x / 1 = 1 / (x - 1), which yields x = (1 + √5) / 2.  After removing the red and green squares we have one of the two scenarios shown, the one on the left occurring when 2 - x < x - 1, the one on the right in the other case.  The first leads to the equation x / 1 = (x - 1) / (2 - x), and x = (1 + √5) / 2, which is not the case.  The one on the right leads to x / 1 = (2 - x) / (x - 1) and our desired solution x = √2.   You are visitor number 1197 to this page.