Puzzle 133 (Corral) [Toroidal]

This is a Friday Corral puzzle, with a twist. The edges of the grid wrap around to each other. The solution is a single connected loop that may cross over the edges and that partitions the grid into “walls” and “empty spaces” (I would not say inside and outside). Note that all clues in this puzzle are finite, so there is no clue that can see a whole row or column (and thus see infinitely far).

Puzzle 133

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8 Responses to “Puzzle 133 (Corral) [Toroidal]”

1. Alan Curry Says:

Should I assume that, whatever you want to call the 2 regions on opposite sides of the partitioning loop, all the clues are still in the same region?

2. MellowMelon Says:

Of course. It would be silly if they could see stuff in walls. π

3. Alan Curry Says:

I have seen a Corral with numbers inside and outside the loop before. The basic rules still work just as well. It was harder than a normal Corral.

4. MellowMelon Says:

Yes, it was on the 2009 USPC, and perhaps harder than anything I’ve posted here.

5. Alan Curry Says:

Now that I’ve actually worked on it a little, that was a stupid question I asked. A single loop on a torus doesn’t partition it at all! I learned that on the toroidal Creek (which was a great puzzle by the way). Not sure how that’s going to affect this one. Still working on it.

6. Alan Curry Says:

Hmm… sometimes when you drawn a loop it partitions the torus… and sometimes it doesn’t. Puzzle complete but I think I failed to learn the topology lesson.

7. MellowMelon Says:

Yes, that is a property of the torus that it does not share with the plane. My favorite way of making this precise is with fundamental groups / homotopy classes. ( http://en.wikipedia.org/wiki/Fundamental_group )

8. rob Says:

Great puzzle.

It seems that the loop will partition the torus if its homotopy class has even “degree”.

In the puzzle, the walls form a simple connected subspace, while the rest has two independent loops (it’s homotopy equivalent to the graph with vertices 5, 6, 13 and two different edges 5-6 as well as single edges 6-13 and 5-13). Does it have the same fundamental group as the torus?

It should be possible to relate the homotopy groups of the two areas and the homotopy group of the loop with Seifert-van Kampen…

How about a toroidal Corral where neither the wall nor the outer cells are simply connected? π