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This one is tougher.

Start with a square. Suppose the top and bottom of the square can be either straight or have an interlocking pattern, as shown in the two examples below:

enter image description here

And suppose the left and right of the square can be either straight, concave or convex, as shown in the examples below:

enter image description here

That gives $2 \times 3 \times 2 \times 3 = 36$ possible squares.

Is it possible to create a $6 \times 6$ jigsaw puzzle (outside borders straight) with these $36$ pieces? Rotation or flipping of pieces is not allowed.

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  • $\begingroup$ Does each piece have to be different from all the others? Otherwise you could use 36 plain squares. $\endgroup$ – Spencer Feb 26 '20 at 22:40
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    $\begingroup$ Sounds like you get one of each of the 36 pieces exactly, without rotation. $\endgroup$ – Joel Rondeau Feb 26 '20 at 22:45
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    $\begingroup$ @JoelRondeau some of the pieces are 180 degree rotations of each other. $\endgroup$ – Spencer Feb 26 '20 at 23:26
  • $\begingroup$ @Spencer Each piece is different from the others. $\endgroup$ – Jens Feb 26 '20 at 23:29
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A 4x9 rectangle is easy to create in a similar way to the previous one.

 +---+---+---+---+---+---+---+---+---+
 |   |   >   >   <   <   >   |   <   |
 +-N-+-N-+-N-+-N-+-N-+-N-+-N-+-N-+-N-+
 |   |   >   >   <   <   >   |   <   |
 +-N-+-N-+-N-+-N-+-N-+-N-+-N-+-N-+-N-+
 |   |   >   >   <   <   >   |   <   |
 +---+---+---+---+---+---+---+---+---+
 |   |   >   >   <   <   >   |   <   |
 +---+---+---+---+---+---+---+---+---+

Each horizontal/vertical line through the puzzle uses the same edge shape throughout. The 5 horizontal lines are chosen such that the 4 adjacent pairs of lines have all 4 possible combinations of top/bottom edge shapes. Similarly the 10 vertical lines are chosen such that the 9 adjacent pairs of lines have all 9 possible combinations of left/right edge shapes. This therefore creates all 4*9=36 tiles.

Using that as a starting point, I rearranged them into a 6x6 square.

  +---+---+---+---+---+---+
  |   >   >   <   <   >   |
  +-N-+-N-+-N-+-N-+-N-+-N-+
  |   >   >   <   <   >   |
  +-N-+-N-+-N-+-N-+-N-+-N-+
  |   >   >   <   |   <   |
  +---+---+---+---+-N-+-N-+
  |   |   <   |   <   >   |
  +-N-+-N-+-N-+---+---+---+
  |   |   <   <   <   |   |
  +-N-+---+---+---+---+---+
  |   |   >   >   <   >   |
  +---+---+---+---+---+---+
I had to break up the 3x2 block, exchanging some pieces with the large 3x6 block, and then it fairly soon all fell into place.

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  • $\begingroup$ Very nicely done! $\endgroup$ – Jens Feb 27 '20 at 14:41
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So, I came up to a organised arrangement for the outermost frame in which I avoid using straights lines where unecessary. turns out I haven't placed the square yet. On the reduce set of possibles I ended up on, it seems impossible

enter image description here

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    $\begingroup$ Welcome to Puzzling! This doesn't seem like a particularly helpful answer - generally for questions like this, we look for either some sort of proof that it's impossible or at least some reason to believe it is. Right now it just seems like you're saying this one particular method doesn't work, and there seem to be many possible methods. $\endgroup$ – Deusovi♦ Feb 26 '20 at 23:17

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