I have several free games on my website, that I find entertaining. One of them has been pissing me off lately, though, because I can’t solve it. I don’t even remember if I ever have solved it in the past, I just know that I can’t seem to be able to now.
It’s a simple premise, but I can’t get it. Can you, and then tell me how you got there?
Solved algorithmically:
Assuming the columns are marked A, B, C, D, E and the rows are marked 1, 2, 3, 4, 5, the solution is as follows:
B1, C1, E1, B2, C2, D2, C3, D3, E3, A4, B4, D4, E4, A5, B5
Well, that light one is sure a head scratcher.
Five years you say? I have one that a substitute teacher assigned us in third grade. She never told us the answer, and I’ve never been able to solve it.
It’s a little visual, so I’ll describe it as well as I can. Picture a sheet of paper. There are three little houses drawn in a row. Beneath the houses, a couple of spaces down, are drawn three little squares representing a power plant, a natural gas plant, and a water pumping station. That’s one little square (plant) drawn beneath each house.
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?
Ultrafilter’s solution worked just fine.
Yeah, the gotcha is that it’s impossible to solve; in mathematical jargon, the graph K[sub]3, 3[/sub] is non-planar.
It’s been on my site for five years. I haven’t been trying to solve it for five years. I generally forget it’s there for years at a time until I stumble across it and give it a go for a few minutes before remembering “oh yeah, this was hard” and giving up.
Hooray! Not sure if I would ever have had the patience to figure that out or not. Thank youuuuuuu.
Interestingly, it doesn’t matter what order you click the squares.
Right: each click on a particular button just says “Flip the lights of these particular squares”; re-ordering the clicks doesn’t change the total number of times any square gets flipped.
(And, of course, we have the added nicety that clicking any square twice is as good as not clicking it at all, since double flipping cancels out.)
I was lazy before, but let me explain more accessibly why the pipe puzzle is impossible to solve:
As you go from house to plant to another house to another plant to the last house to the last plant and then back to the first house, you draw a loop out of six pipes. If the loop doesn’t intersect itself, it acts as a border separating the Earth into two regions: the “inside” and “outside” of the loop.
That leaves three pipes to place; avoiding intersections means each time you place one of these pipes, you place it entirely within an existing region (splitting that region into two). Thus, after laying all the pipes, we’d be up to 5 regions on the Earth, separated by the pipe-borders.
But!
Each region’s border contains an even number of pipes (since pipes alternate between houses and plants). And the number can’t be just two (in that case, the second pipe would just be going between the same buildings as the first pipe).
So each region’s border has at least 4 pipes.
Since we have 5 regions, and each pipe borders 2 regions, that means we must have at least 5 * 4 / 2 = 10 pipes [to pull this off without intersections].
But we don’t. We only have 9 pipes. And so it’s impossible to pull this off without intersections.
It’s not impossible, though, because the houses are not points. It never says you can’t go under another house.
I’m very surprised that ultrafilter’s solution is not symmetric-- Usually the solutions to this sort of puzzle are. I’d expect that for each room, there’d be a set of switchings which would toggle exactly that room, and that a total solution could be obtained just by XORing all of those sets together, which would (or at least could) lead to a symmetric solution.
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.
Er, the symmetric such actions, of course.
Actually, the same invariant is also preserved even if one only demands symmetry under orthogonal reflection and not diagonal reflection. So one cannot find a solution with even that weaker form of symmetry.
ignore this post
Consider it ignored.
Moved MPSIMS --> the Game Room.
Isn’t that violating the spirit of the puzzle?
When I learned this I was told that you can’t do that.
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.