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Most Correct Signposts? On a Torus?

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kothmann:
I saw @KBellon's I Love the Postman image today and noticed that there are 9 correct signposts in the arrangement of the unpunched tiles:

Note that there are also 2 incompatible edges on the RRRR tile.   >:(

I thought it might be fun to determine the highest number of correct signposts that could be achieved using only these tiles with no incompatible edges.  Notice that there are an odd number of road edges, so you can't make one closed loop.   :(

As I was writing this, I remembered a super-fun Tower (Exp#4) puzzle which uses all 18 Tower tiles to make a 6x3 tessellation of a torus!  See this animation for an example of how a rectangle can represent the surface of a torus.

So, the original question is now 5 questions:  >:D

* What is the highest number of correct signposts in a planar arrangement of these 12 tiles with no incompatible edges, but no other constraints on the layout?
* Are there 4x3 or 6x2 rectangular arrangements of these tiles with no incompatible edges on the plane?
* If the answer to #2 is "yes", what is the highest number of correct signposts in such an arrangement?
* Are there 4x3 or 6x2 rectangular arrangements of these tiles with no incompatible edge on a torus?
* If the answer to #4 is "yes", what is the highest number of correct signposts in an arrangement on a torus?

Linkback: https://www.carcassonnecentral.com/community/index.php?topic=5584.0

kothmann:
Ugh I’m so foolish.  :o

Of course there is no torus, because the unpaired road always makes an incompatible edge on a torus.   :-[

So, just questions 1, 2, and 3.    ???

DrMeeple:

--- Quote from: kothmann on December 06, 2021, 08:30:55 PM ---Ugh I’m so foolish.  :o

Of course there is no torus, because the unpaired road always makes an incompatible edge on a torus.   :-[

So, just questions 1, 2, and 3.    ???

--- End quote ---



Sent from my iPhone using Tapatalk

kothmann:
1. My best non-rectangular arrangement is 20 correct signs
2. Yes both 6x2 and 4x3 are possible.
3. My bests are 13 correct signs for 6x2 and 10 correct signs for 4x3.

I don’t have the tiles so my answers were constructed using post-it notes.   ::)

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