TITLE: Uniqueness of spanning tree on a grid.
QUESTION [8 upvotes]: 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.)

REPLY [9 votes]: No, solutions are not unique. The four "T" shaped pieces in the grid below can be rotated into either of two configurations:
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