5room puzzle variants | diagramwith 16 openings/doors
| 3Drendering of the puzzle
| Thisold popular puzzle, called “Five Room Housepuzzle” (also known as “Wallsand Lines puzzle”, or “Crossthe Network puzzle”), is canonically representedas a rectangular diagram divided into five rooms, asshown opposite. The object of the puzzle is to draw acontinuous path through the walls of all 5 rooms, withoutgoing through any wall twice, and without crossing anypath. The path can, of course, end in any room, not necessarilyin the room from where it started. Some puzzle diagramsrepresent the rooms with openings supposed to be doors.In this instance, the challenge is to visit every roomof the apartment by walking through every door exactlyonce.Requirementsfor solvability Whether starting and ending in the same room, or startingin one room and ending in another one, every otherroom of the diagram/apartment must have an even numberof doors... That is, pair(s) of ‘in’ and ‘out’ doors (asdoors CANNOT be used TWICE, we have then to use aneven number of doors as we ENTER and LEAVE those rooms). Let’ssuppose we start in a room with an odd number ofdoors, then it is possible to visit all the 5 roomsof the apartment if and ONLY if another room hasan odd number of doors - representing thedeparture and the arrival points of the continuouspath - , and all the other rooms have aneven number of doors. In a few words, for this topologicalpuzzle to be solvable, there may NOT be more thanTWO rooms with an odd number of doors. Since thepuzzle has THREE rooms with an odd number of openings/doors,it is mathematically impossible to complete a circuitcrossing. Analogously,a continuous line that enters and leaves one of therooms crosses two walls. Since the THREE contiguouslarger rooms each have an odd number of walls tobe crossed, it follows that an END of a line mustbe inside each of them if all the 16 walls are crossed.But a unicursal line has only TWO ends, thiscontradiction makes the 5 Room House puzzle unsolvable. However,if we close a door or add an extra room to the puzzle(see fig. a and b below), thenit becomes solvable. Now, you can easily draw onecontinuous line that passes through every openingexactly once... Try it! (the five-room variant onthe left (fig. a) is just a little harder to solve,because you have to figure out where to start) TheFive Room House is actually a classic example ofan impossible puzzle — one that bears no positivesolution. In this particular case, the solutionconsists in finding that the problem has no solution!(remember: puzzles always have one, several or no solutions;see tipsto puzzle solving) Graphtheory Theinsolubility of the 5 Room House problem can be provedusing a graphtheory approach, with each room being a vertex andeach wall being an edge of the graph (seeimage opposite). In fact, this puzzle is similar tothe famous “sevenbridges of Königsberg” problem thanksto which the eminent Swiss mathematician LeonhardEuler laid the foundations of graph theory. Euler wondered whether there was a way of traversingeach of the 7 bridges over the river Pregel at Königsberg (nowKaliningrad) once and only once, starting and returningat the same point in the town. He finally realizedthat the problem had no solutions! |
Tricksto 'solve' the puzzle As you experienced, this puzzle is impossible to solveon paper... But ‘impossible’ puzzles sometimeshave out-of-the-box solutions, as the non-standardsolution depicted below.
Anotherneat out-of-the box solution... Everything down to this point has been in 2 dimensions,either a diagram drawn on paper, or a five room apartmenton a flat surface. In order to draw a continuous paththat goes from one room to another without crossinga line or going through a door twice, you have to reproducethe 5 room house puzzle onto a surface that is nottopologically equivalent to a sheet of paper. The solidthat may help you is a torus, a kind of ring-shapedsolid resembling a doughnut or a bagel. The puzzlediagram should be reproduced so that the hole of thetorus is inside one of the 3 larger rooms, as shownin the example below.
Allthe Most Wanted Puzzle Solutions in a look! |