Uniqueness of spanning tree on a grid.












2












$begingroup$


When I was at the Graduate Student Combinatorics Conference earlier this month, someone introduced me to a puzzle game called Noodles!.



The game starts with a collection of "pipes" on a grid (centered on each vertex), clicking on a piece rotates it $90^circ$, and a piece can be rotated any number of times. The goal is to turn the final configuration of pipes into a spanning tree (of the grid graph), as shown in the screenshots below.



Example



An example of Noodles!



Question



We left the conference with an unsolved question:
Are solutions to this puzzle always unique? Or is it possible to come up with a starting configuration (on any size grid) that has multiple trees as solutions?



(The prevailing guess is that solutions are unique, but nobody could manage to prove it.)






combinatorics graph-theory puzzle trees







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    This genre of logic puzzle is known originally as Netwalk.
    $endgroup$
    – Mike Earnest
    4 hours ago










  • $begingroup$
    @MikeEarnest, thanks for the reference. Based on this picture from this Reddit thread, it looks like (some versions of) Netwalk are on a torus instead of a grid—although that presents an interesting generalization of this question.
    $endgroup$
    – Peter Kagey
    4 hours ago


















2












$begingroup$


When I was at the Graduate Student Combinatorics Conference earlier this month, someone introduced me to a puzzle game called Noodles!.



The game starts with a collection of "pipes" on a grid (centered on each vertex), clicking on a piece rotates it $90^circ$, and a piece can be rotated any number of times. The goal is to turn the final configuration of pipes into a spanning tree (of the grid graph), as shown in the screenshots below.



Example



An example of Noodles!



Question



We left the conference with an unsolved question:
Are solutions to this puzzle always unique? Or is it possible to come up with a starting configuration (on any size grid) that has multiple trees as solutions?



(The prevailing guess is that solutions are unique, but nobody could manage to prove it.)






combinatorics graph-theory puzzle trees







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    This genre of logic puzzle is known originally as Netwalk.
    $endgroup$
    – Mike Earnest
    4 hours ago










  • $begingroup$
    @MikeEarnest, thanks for the reference. Based on this picture from this Reddit thread, it looks like (some versions of) Netwalk are on a torus instead of a grid—although that presents an interesting generalization of this question.
    $endgroup$
    – Peter Kagey
    4 hours ago
















2












2








2


1



$begingroup$


When I was at the Graduate Student Combinatorics Conference earlier this month, someone introduced me to a puzzle game called Noodles!.



The game starts with a collection of "pipes" on a grid (centered on each vertex), clicking on a piece rotates it $90^circ$, and a piece can be rotated any number of times. The goal is to turn the final configuration of pipes into a spanning tree (of the grid graph), as shown in the screenshots below.



Example



An example of Noodles!



Question



We left the conference with an unsolved question:
Are solutions to this puzzle always unique? Or is it possible to come up with a starting configuration (on any size grid) that has multiple trees as solutions?



(The prevailing guess is that solutions are unique, but nobody could manage to prove it.)






combinatorics graph-theory puzzle trees







share|cite|improve this question









$endgroup$




When I was at the Graduate Student Combinatorics Conference earlier this month, someone introduced me to a puzzle game called Noodles!.



The game starts with a collection of "pipes" on a grid (centered on each vertex), clicking on a piece rotates it $90^circ$, and a piece can be rotated any number of times. The goal is to turn the final configuration of pipes into a spanning tree (of the grid graph), as shown in the screenshots below.



Example



An example of Noodles!



Question



We left the conference with an unsolved question:
Are solutions to this puzzle always unique? Or is it possible to come up with a starting configuration (on any size grid) that has multiple trees as solutions?



(The prevailing guess is that solutions are unique, but nobody could manage to prove it.)






combinatorics graph-theory puzzle trees




combinatorics graph-theory puzzle trees






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 4 hours ago









Peter KageyPeter Kagey

1,61272053




1,61272053








  • 1




    $begingroup$
    This genre of logic puzzle is known originally as Netwalk.
    $endgroup$
    – Mike Earnest
    4 hours ago










  • $begingroup$
    @MikeEarnest, thanks for the reference. Based on this picture from this Reddit thread, it looks like (some versions of) Netwalk are on a torus instead of a grid—although that presents an interesting generalization of this question.
    $endgroup$
    – Peter Kagey
    4 hours ago
















  • 1




    $begingroup$
    This genre of logic puzzle is known originally as Netwalk.
    $endgroup$
    – Mike Earnest
    4 hours ago










  • $begingroup$
    @MikeEarnest, thanks for the reference. Based on this picture from this Reddit thread, it looks like (some versions of) Netwalk are on a torus instead of a grid—although that presents an interesting generalization of this question.
    $endgroup$
    – Peter Kagey
    4 hours ago










1




1




$begingroup$
This genre of logic puzzle is known originally as Netwalk.
$endgroup$
– Mike Earnest
4 hours ago




$begingroup$
This genre of logic puzzle is known originally as Netwalk.
$endgroup$
– Mike Earnest
4 hours ago












$begingroup$
@MikeEarnest, thanks for the reference. Based on this picture from this Reddit thread, it looks like (some versions of) Netwalk are on a torus instead of a grid—although that presents an interesting generalization of this question.
$endgroup$
– Peter Kagey
4 hours ago






$begingroup$
@MikeEarnest, thanks for the reference. Based on this picture from this Reddit thread, it looks like (some versions of) Netwalk are on a torus instead of a grid—although that presents an interesting generalization of this question.
$endgroup$
– Peter Kagey
4 hours ago












1 Answer
1






active

oldest

votes


















4












$begingroup$

No, solutions are not unique. The four "T" shaped pieces in the grid below can be rotated into either of two configurations:



┏━━━┓

┗┓╻╻┃

╺┫┣┛┃

┏┫┣╸┃

╹╹┗━┛

┏━━━┓

┗┓╻╻┃

╺┻┻┛┃

┏┳┳╸┃

╹╹┗━┛





share|cite|improve this answer








New contributor




edderiofer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$













  • $begingroup$
    Based on the picture in the question, all three ends of the T shapes have to connect to an adjacent piece; you can't have one end of the T run directly into the flat side of a | piece.
    $endgroup$
    – Misha Lavrov
    3 hours ago










  • $begingroup$
    It doesn't "run directly into the flat side of a | piece". There's actually a dead end "╸" piece in between, so this satisfies all the rules.
    $endgroup$
    – edderiofer
    2 hours ago










  • $begingroup$
    Oh, I see. I couldn't quite read it in your answer, but it showed up when I highlighted it and now everything makes sense.
    $endgroup$
    – Misha Lavrov
    2 hours ago






  • 1




    $begingroup$
    Here is an interactive illustration of the solution: chiark.greenend.org.uk/~sgtatham/puzzles/js/…
    $endgroup$
    – Mike Earnest
    2 hours ago












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1 Answer
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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes









4












$begingroup$

No, solutions are not unique. The four "T" shaped pieces in the grid below can be rotated into either of two configurations:



┏━━━┓

┗┓╻╻┃

╺┫┣┛┃

┏┫┣╸┃

╹╹┗━┛

┏━━━┓

┗┓╻╻┃

╺┻┻┛┃

┏┳┳╸┃

╹╹┗━┛





share|cite|improve this answer








New contributor




edderiofer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$













  • $begingroup$
    Based on the picture in the question, all three ends of the T shapes have to connect to an adjacent piece; you can't have one end of the T run directly into the flat side of a | piece.
    $endgroup$
    – Misha Lavrov
    3 hours ago










  • $begingroup$
    It doesn't "run directly into the flat side of a | piece". There's actually a dead end "╸" piece in between, so this satisfies all the rules.
    $endgroup$
    – edderiofer
    2 hours ago










  • $begingroup$
    Oh, I see. I couldn't quite read it in your answer, but it showed up when I highlighted it and now everything makes sense.
    $endgroup$
    – Misha Lavrov
    2 hours ago






  • 1




    $begingroup$
    Here is an interactive illustration of the solution: chiark.greenend.org.uk/~sgtatham/puzzles/js/…
    $endgroup$
    – Mike Earnest
    2 hours ago
















4












$begingroup$

No, solutions are not unique. The four "T" shaped pieces in the grid below can be rotated into either of two configurations:



┏━━━┓

┗┓╻╻┃

╺┫┣┛┃

┏┫┣╸┃

╹╹┗━┛

┏━━━┓

┗┓╻╻┃

╺┻┻┛┃

┏┳┳╸┃

╹╹┗━┛





share|cite|improve this answer








New contributor




edderiofer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$













  • $begingroup$
    Based on the picture in the question, all three ends of the T shapes have to connect to an adjacent piece; you can't have one end of the T run directly into the flat side of a | piece.
    $endgroup$
    – Misha Lavrov
    3 hours ago










  • $begingroup$
    It doesn't "run directly into the flat side of a | piece". There's actually a dead end "╸" piece in between, so this satisfies all the rules.
    $endgroup$
    – edderiofer
    2 hours ago










  • $begingroup$
    Oh, I see. I couldn't quite read it in your answer, but it showed up when I highlighted it and now everything makes sense.
    $endgroup$
    – Misha Lavrov
    2 hours ago






  • 1




    $begingroup$
    Here is an interactive illustration of the solution: chiark.greenend.org.uk/~sgtatham/puzzles/js/…
    $endgroup$
    – Mike Earnest
    2 hours ago














4












4








4





$begingroup$

No, solutions are not unique. The four "T" shaped pieces in the grid below can be rotated into either of two configurations:



┏━━━┓

┗┓╻╻┃

╺┫┣┛┃

┏┫┣╸┃

╹╹┗━┛

┏━━━┓

┗┓╻╻┃

╺┻┻┛┃

┏┳┳╸┃

╹╹┗━┛





share|cite|improve this answer








New contributor




edderiofer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$



No, solutions are not unique. The four "T" shaped pieces in the grid below can be rotated into either of two configurations:



┏━━━┓

┗┓╻╻┃

╺┫┣┛┃

┏┫┣╸┃

╹╹┗━┛

┏━━━┓

┗┓╻╻┃

╺┻┻┛┃

┏┳┳╸┃

╹╹┗━┛






share|cite|improve this answer








New contributor




edderiofer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this answer



share|cite|improve this answer






New contributor




edderiofer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









answered 3 hours ago









edderioferedderiofer

1561




1561




New contributor




edderiofer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





edderiofer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






edderiofer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












  • $begingroup$
    Based on the picture in the question, all three ends of the T shapes have to connect to an adjacent piece; you can't have one end of the T run directly into the flat side of a | piece.
    $endgroup$
    – Misha Lavrov
    3 hours ago










  • $begingroup$
    It doesn't "run directly into the flat side of a | piece". There's actually a dead end "╸" piece in between, so this satisfies all the rules.
    $endgroup$
    – edderiofer
    2 hours ago










  • $begingroup$
    Oh, I see. I couldn't quite read it in your answer, but it showed up when I highlighted it and now everything makes sense.
    $endgroup$
    – Misha Lavrov
    2 hours ago






  • 1




    $begingroup$
    Here is an interactive illustration of the solution: chiark.greenend.org.uk/~sgtatham/puzzles/js/…
    $endgroup$
    – Mike Earnest
    2 hours ago


















  • $begingroup$
    Based on the picture in the question, all three ends of the T shapes have to connect to an adjacent piece; you can't have one end of the T run directly into the flat side of a | piece.
    $endgroup$
    – Misha Lavrov
    3 hours ago










  • $begingroup$
    It doesn't "run directly into the flat side of a | piece". There's actually a dead end "╸" piece in between, so this satisfies all the rules.
    $endgroup$
    – edderiofer
    2 hours ago










  • $begingroup$
    Oh, I see. I couldn't quite read it in your answer, but it showed up when I highlighted it and now everything makes sense.
    $endgroup$
    – Misha Lavrov
    2 hours ago






  • 1




    $begingroup$
    Here is an interactive illustration of the solution: chiark.greenend.org.uk/~sgtatham/puzzles/js/…
    $endgroup$
    – Mike Earnest
    2 hours ago
















$begingroup$
Based on the picture in the question, all three ends of the T shapes have to connect to an adjacent piece; you can't have one end of the T run directly into the flat side of a | piece.
$endgroup$
– Misha Lavrov
3 hours ago




$begingroup$
Based on the picture in the question, all three ends of the T shapes have to connect to an adjacent piece; you can't have one end of the T run directly into the flat side of a | piece.
$endgroup$
– Misha Lavrov
3 hours ago












$begingroup$
It doesn't "run directly into the flat side of a | piece". There's actually a dead end "╸" piece in between, so this satisfies all the rules.
$endgroup$
– edderiofer
2 hours ago




$begingroup$
It doesn't "run directly into the flat side of a | piece". There's actually a dead end "╸" piece in between, so this satisfies all the rules.
$endgroup$
– edderiofer
2 hours ago












$begingroup$
Oh, I see. I couldn't quite read it in your answer, but it showed up when I highlighted it and now everything makes sense.
$endgroup$
– Misha Lavrov
2 hours ago




$begingroup$
Oh, I see. I couldn't quite read it in your answer, but it showed up when I highlighted it and now everything makes sense.
$endgroup$
– Misha Lavrov
2 hours ago




1




1




$begingroup$
Here is an interactive illustration of the solution: chiark.greenend.org.uk/~sgtatham/puzzles/js/…
$endgroup$
– Mike Earnest
2 hours ago




$begingroup$
Here is an interactive illustration of the solution: chiark.greenend.org.uk/~sgtatham/puzzles/js/…
$endgroup$
– Mike Earnest
2 hours ago


















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