# I can't solve this puzzle that's been on my own website for at least 5 years **URL:** \ **Category:** The Game Room\ **Created:** [August 22, 2011, 1:55am UTC](https://boards.straightdope.com/t/i-cant-solve-this-puzzle-thats-been-on-my-own-website-for-at-least-5-years/593603 "2011-08-22T01:55:16Z")\ **Posts on this page:** 16\ **Page:** 2
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [August 22, 2011, 5:05pm UTC](https://boards.straightdope.com/t/i-cant-solve-this-puzzle-thats-been-on-my-own-website-for-at-least-5-years/593603/21 "2011-08-22T17:05:18Z")
That sort of reminds me of my solution to the “nine dots, four lines” puzzle, which is usually presented in such a way that it can actually be solved in three lines. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 22, 2011, 5:43pm UTC](https://boards.straightdope.com/t/i-cant-solve-this-puzzle-thats-been-on-my-own-website-for-at-least-5-years/593603/22 "2011-08-22T17:43:31Z")
> [@Indistinguishable](#): > > Nope: a symmetric solution can’t individually toggle the center (if symmetry includes reflection across the orthogonal and diagonal axes). There’s six kinds of buttons, up to such symmetry, but their actions only span a five-dimensional space. Specifically, their span is the actions which preserve parity along C1, C2, C3. This of course means it’s impossible to symmetrically solve the puzzle. The solution to the equivalent 7 x 7 puzzle is symmetric any way you could like (see below). What’s the difference between that and the 5 x 5 case? A1, B1, D1, F1, G1 A2, B2, C2, E2, F2, G2 B3, C3, E3, F3 A4, D4, G4 B5, C5, E5, F5 A6, B6, C6, E6, F6, G6 A7, B7, D7, F7, G7 (I’m using the solver [here](http://www.ueda.info.waseda.ac.jp/~n-kato/lightsout/index.html). The largest puzzle they can solve is 7 x 7, so if you want a larger puzzle solved, it’ll take a little while.) ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 22, 2011, 5:57pm UTC](https://boards.straightdope.com/t/i-cant-solve-this-puzzle-thats-been-on-my-own-website-for-at-least-5-years/593603/23 "2011-08-22T17:57:42Z")
In the 7 by 7 case, the buttons are all linearly independent, so one must have a solution for every configuration and also a symmetric solution for every symmetric configuration. The number of dimensions of possible configurations equals the number of dimensions of possible actions. In the 5 by 5 case, the buttons are not all linearly independent, so not every configuration has a solution; indeed, even symmetric combinations of buttons can demonstrate linear dependence (e.g., clicking every button not on one of the large diagonals is the same as doing nothing), so not every symmetric configuration has a symmetric solution. The number of dimensions of possible configurations exceeds the number of dimensions of possible actions. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 22, 2011, 6:29pm UTC](https://boards.straightdope.com/t/i-cant-solve-this-puzzle-thats-been-on-my-own-website-for-at-least-5-years/593603/24 "2011-08-22T18:29:04Z")
> [@Indistinguishable](#): > > The number of dimensions of possible configurations equals the number of dimensions of possible actions. Just to be more explicit as to why this entails symmetric solutions to symmetric configurations: This means each configuration has a unique solution, and thus the unique solution to a configuration must in itself contain all the same symmetries as the configuration. (That is, the map sending button combinations to their cumulative actions always sends symmetric button combinations to symmetric actions; in the 7 by 7 case, this map is a bijection, so we have conversely that every symmetric action comes from a symmetric button combination) ---
**Author:** ![davidm](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/davidm/32/225_2.png) [@davidm](https://boards.straightdope.com/u/davidm)\ **Post date:** [August 22, 2011, 6:33pm UTC](https://boards.straightdope.com/t/i-cant-solve-this-puzzle-thats-been-on-my-own-website-for-at-least-5-years/593603/25 "2011-08-22T18:33:04Z")
> [@Hoopy\_Frood](#): > > How can you violate the spirit of a puzzle that isn’t solveable? > > An unsolveable puzzle is in of itself a violation of the spirit of puzzles to begin with…since, ya know the point is to solve them. > > Unless the real point is to determine it’s unsolveable, but most people who haven’t studied graph theory aren’t going to realize it. Heck, I have studied some graph theory (though, I never use it, and hated it when I had to study it) and I wouldn’t have figured that puzzle unsolveable if you treat each plant/house as a vertex just by looking/hearing about it. When I learned it, the lesson was that it was unsolvable. That was shown only after we had spent time trying to solve it. One of the rules was that a line couldn’t go under a house or a utility. Obviously that’s one solution if the puzzle is presented without that limitation, but I think the whole point of the exercise is to show that some things are unsolvable and can be proven so. ---
**Author:** ![cmkeller](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cmkeller/32/118_2.png) [@cmkeller](https://boards.straightdope.com/u/cmkeller)\ **Post date:** [August 22, 2011, 7:01pm UTC](https://boards.straightdope.com/t/i-cant-solve-this-puzzle-thats-been-on-my-own-website-for-at-least-5-years/593603/26 "2011-08-22T19:01:13Z")
Well, I solved it but darned if I used a system. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 22, 2011, 7:16pm UTC](https://boards.straightdope.com/t/i-cant-solve-this-puzzle-thats-been-on-my-own-website-for-at-least-5-years/593603/27 "2011-08-22T19:16:08Z")
One point which hasn’t been remarked upon, incidentally, is that **ultrafilter** ’s solution does have _one_ symmetry (reflection across the diagonal from E1 to A5). ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 22, 2011, 7:55pm UTC](https://boards.straightdope.com/t/i-cant-solve-this-puzzle-thats-been-on-my-own-website-for-at-least-5-years/593603/28 "2011-08-22T19:55:57Z")
With regard to the houses/utility puzzle, this is a good lesson in how different models give different results. To recap: [ol] [li]If you model the houses and utilities as points in a plane, then there’s no solution as described above.[/li][li]If you model the houses and utilities as sets of nonzero area in the plane, there is a solution, but you have to run one line through a house or utility that it’s not serving to get it.[/li][li]If, on the other hand, your model incorporates a third dimension, the problem is trivial: simply run one utility cable under another.[/li][/ol] ---
**Author:** ![Munch](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/munch/32/5281_2.png) [@Munch](https://boards.straightdope.com/u/Munch)\ **Post date:** [August 22, 2011, 10:46pm UTC](https://boards.straightdope.com/t/i-cant-solve-this-puzzle-thats-been-on-my-own-website-for-at-least-5-years/593603/29 "2011-08-22T22:46:55Z")
> [@Two\_Many\_Cats](#): > > The object is to use a pen to connect up the three houses with all three plants drawing three pipelines from each plant, one to each house. The pipelines cannot cross each other. > > I’ve spent hours in waiting rooms years ago trying to figure this one out and never have. Has anyone else seen this problem? Does anyone know the solution? Take the piece of paper, and roll it into a tube. Now draw your final line. ---
**Author:** ![Two\_Many\_Cats](https://avatars.discourse-cdn.com/v4/letter/t/8dc957/32.png) [@Two\_Many\_Cats](https://boards.straightdope.com/u/Two_Many_Cats)\ **Post date:** [August 22, 2011, 11:07pm UTC](https://boards.straightdope.com/t/i-cant-solve-this-puzzle-thats-been-on-my-own-website-for-at-least-5-years/593603/30 "2011-08-22T23:07:10Z")
> [@davidm](#): > > Obviously that’s one solution if the puzzle is presented without that limitation, but I think the whole point of the exercise is to show that some things are unsolvable and can be proven so. I think the point of this exercise was to get a third grade class to sit down and be quiet which is the goal of any substitute teacher. The thing that kept me at it was that some of my classmates claimed to have solved it, but they would never tell me how. So another point of this exercise was to prove that some of my classmates were lying sacks of shit, which I had always suspected. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 23, 2011, 2:51am UTC](https://boards.straightdope.com/t/i-cant-solve-this-puzzle-thats-been-on-my-own-website-for-at-least-5-years/593603/31 "2011-08-23T02:51:52Z")
> [@Munch](#): > > Take the piece of paper, and roll it into a tube. Now draw your final line. This doesn’t help… if you could draw it on the sides of a cylinder, you could draw it on the surface of the Earth, which, as above noted, is impossible. Put another way, all folding the paper into a tube lets you do for the final line is have it exit out one side of the paper and re-enter on the opposite side. But you can already accomplish the same thing without having to fold the paper: once you get to the one side of the paper, just run around the margins till you make it to the other side. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [August 23, 2011, 3:16am UTC](https://boards.straightdope.com/t/i-cant-solve-this-puzzle-thats-been-on-my-own-website-for-at-least-5-years/593603/32 "2011-08-23T03:16:53Z")
Or, looking at it another way: Stretch the bottom circle of your cylinder out, and maybe shrink the top circle, until it’s a flat disk with a hole in the middle. If you can solve it on a flat piece of paper with a hole, then surely you can solve it on one without a hole. Now, if you also attach the ends of your cylinder to make a torus, that might help. Oh, and I figured out why my intuition was that the lights-on puzzle should be symmetrical. The existence of an asymmetrical solution implies the existence of many identities (sets of switches you can flip to leave the state the same), and I wasn’t expecting any nontrivial identities. Take the original solution, rotate it in some way in which it doesn’t have a symmetry, and XOR the two together, and you’ll get an identity. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 23, 2011, 3:33am UTC](https://boards.straightdope.com/t/i-cant-solve-this-puzzle-thats-been-on-my-own-website-for-at-least-5-years/593603/33 "2011-08-23T03:33:18Z")
Right; the existence of nontrivial identities is what I meant by saying the buttons’ actions weren’t linearly independent (as vectors in the 5 by 5 dimensional space over the field of integers modulo 2). As you note, if you have linear independence, the unique solution to any particular starting configuration must itself have all the same symmetries as that configuration. ---
**Author:** ![Hal\_Briston](https://avatars.discourse-cdn.com/v4/letter/h/a9a28c/32.png) [@Hal\_Briston](https://boards.straightdope.com/u/Hal_Briston)\ **Post date:** [August 23, 2011, 2:46pm UTC](https://boards.straightdope.com/t/i-cant-solve-this-puzzle-thats-been-on-my-own-website-for-at-least-5-years/593603/34 "2011-08-23T14:46:19Z")
The solution is astonishingly simple: click randomly for about 30 seconds without any logic or pattern: voilà! ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [August 23, 2011, 8:39pm UTC](https://boards.straightdope.com/t/i-cant-solve-this-puzzle-thats-been-on-my-own-website-for-at-least-5-years/593603/35 "2011-08-23T20:39:16Z")
Ah, I understand what you’re saying now. Yeah, we’re on the same page. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 23, 2011, 8:58pm UTC](https://boards.straightdope.com/t/i-cant-solve-this-puzzle-thats-been-on-my-own-website-for-at-least-5-years/593603/36 "2011-08-23T20:58:49Z")
> [@Chronos](#): > > Or, looking at it another way: Stretch the bottom circle of your cylinder out, and maybe shrink the top circle, until it’s a flat disk with a hole in the middle. If you can solve it on a flat piece of paper with a hole, then surely you can solve it on one without a hole. Ooh, yeah, that’s a much nicer description. > [@Chronos](#): > > Now, if you also attach the ends of your cylinder to make a torus, that might help. Indeed; [here](http://mathworld.wolfram.com/images/eps-gif/UtilityGraphTorus_300.gif)’s an illustration from MathWorld of how it can be done on a torus. [Previous page](https://boards.straightdope.com/t/i-cant-solve-this-puzzle-thats-been-on-my-own-website-for-at-least-5-years/593603.md?page=1)