As its name suggests, this puzzle consists of a house with five rooms and 16 doors, as depicted in the diagram below. A rope passes through and cannot be removed from the upper middle inner door (marked with a dashed red segment). The objective is to pass the rope once through each and every door without crossing the rope over itself or over any walls. (If the rope could be passed more than once through any door, then the solution would be trivial.)
There is no solution, as is proven here, beginning with a lemma:
Lemma: In any solution, a room with an odd number of doors must contain a rope end.
Proof of Lemma: Let N be the number of doors. N is a positive odd integer (1, 3, 5,...) and can therefore be expressed as N = 1 + (2 * P), where P is a nonnegative integer (0, 1, 2,...). P is the number of pairs of two doors. In any solution, the rope can originate from inside or outside the room. If inside, then the room contains a rope end. If outside, then the rope must enter from the outside through one door and exit out another door. This happens P times, since two doors are used each time. (Of course, if N = 1 then P = 0.) Both ends of the rope are now outside. One of them must be passed into the room to fill the one remaining door. Hence, the rope must end in the room, since it cannot be passed outside through any door a second time, nor over a wall.
Theorem: The puzzle has no solution.
Proof of Theorem: Three of the rooms have an odd number of doors (five). By the Lemma, each of the three rooms must contain a rope end in any solution. But the puzzle uses only one rope, which has only two ends. Thus, there can be no solution to this puzzle.
Postscript: My family and I first encountered this puzzle at the Oregon Museum of Science and Industry (OMSI) in Portland, Oregon during the 1980s. It was one of many exhibits, and the only one labeled as having never been solved. My uncle Robert became quite intrigued by the puzzle, disseminated a formal diagram of it to the family members, and challenged us to either find a solution or prove that there isn't one. I wrote this proof and entitled it "The OMSI Rope Puzzle Untangled", unaware that the puzzle had been presented to the public decades earlier, by Martin Gardner, in his book Scientific American Book of Mathematical Puzzles & Diversions (1957).
Postscript: A Guest Services Floor Supervisor at the OMSI confirmed that, as of 2018-05, three decades later, the puzzle is still on display at the museum! She asked my permission to add a print-out of this article to the puzzle answer guide available at their concierge window.