Uniqueness of spanning tree on a grid.
$begingroup$
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
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
asked 4 hours ago
Peter KageyPeter Kagey
1,61272053
$endgroup$
add a comment |
$begingroup$
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
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
asked 4 hours ago
Peter KageyPeter Kagey
1,61272053
$endgroup$
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
add a comment |
$begingroup$
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
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
asked 4 hours ago
Peter KageyPeter Kagey
1,61272053
$endgroup$
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
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
asked 4 hours ago
Peter KageyPeter Kagey
1,61272053
asked 4 hours ago
Peter KageyPeter Kagey
1,61272053
asked 4 hours ago
Peter KageyPeter Kagey
1,61272053
asked 4 hours ago
Peter KageyPeter Kagey
1,61272053
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
add a comment |
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
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
No, solutions are not unique. The four "T" shaped pieces in the grid below can be rotated into either of two configurations:
┏━━━┓
┗┓╻╻┃
╺┫┣┛┃
┏┫┣╸┃
╹╹┗━┛
┏━━━┓
┗┓╻╻┃
╺┻┻┛┃
┏┳┳╸┃
╹╹┗━┛
answered 3 hours ago
edderioferedderiofer
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.
$endgroup$
$begingroup$
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
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
No, solutions are not unique. The four "T" shaped pieces in the grid below can be rotated into either of two configurations:
┏━━━┓
┗┓╻╻┃
╺┫┣┛┃
┏┫┣╸┃
╹╹┗━┛
┏━━━┓
┗┓╻╻┃
╺┻┻┛┃
┏┳┳╸┃
╹╹┗━┛
answered 3 hours ago
edderioferedderiofer
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.
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
No, solutions are not unique. The four "T" shaped pieces in the grid below can be rotated into either of two configurations:
┏━━━┓
┗┓╻╻┃
╺┫┣┛┃
┏┫┣╸┃
╹╹┗━┛
┏━━━┓
┗┓╻╻┃
╺┻┻┛┃
┏┳┳╸┃
╹╹┗━┛
answered 3 hours ago
edderioferedderiofer
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.
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
No, solutions are not unique. The four "T" shaped pieces in the grid below can be rotated into either of two configurations:
┏━━━┓
┗┓╻╻┃
╺┫┣┛┃
┏┫┣╸┃
╹╹┗━┛
┏━━━┓
┗┓╻╻┃
╺┻┻┛┃
┏┳┳╸┃
╹╹┗━┛
answered 3 hours ago
edderioferedderiofer
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.
$endgroup$
No, solutions are not unique. The four "T" shaped pieces in the grid below can be rotated into either of two configurations:
┏━━━┓
┗┓╻╻┃
╺┫┣┛┃
┏┫┣╸┃
╹╹┗━┛
┏━━━┓
┗┓╻╻┃
╺┻┻┛┃
┏┳┳╸┃
╹╹┗━┛
answered 3 hours ago
edderioferedderiofer
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.
answered 3 hours ago
edderioferedderiofer
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.
answered 3 hours ago
edderioferedderiofer
1561
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
add a comment |
$begingroup$
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
add a comment |
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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