This is a fun puzzle I invented 50 years ago. ("Loops of Zen" is rather similar, idea-wise.) Here is a random solution (doesn't spoil much)...

a valid solution

...and here are the building blocks:

  • Basic is a 3*3 square.
  • Any B/W coloring of it that doesn't contain a 2*2 checkerboard gives a valid piece. (I.e. the thingies marking the corners aren't!)
  • Since only the border between B and W was relevant for me (the B/W setup here is merely more elegant and esthetic), a piece and its negative image are considered the same.
  • Rotated images are considered the same.
  • Mirrored images are considered different.
  • You can count that under these rules there are exactly 45 valid pieces. The one with the looping border goes into the center and counts as "special" (reason implied below). Filling up with four corners (also "special") all pieces fit into a 7*7 square.
  • Puzzle Medium Hard: With corners and center fixed, match all B/W borders. You are allowed to rotate and color-reverse any piece if needed (including the center, but not the corners, i.e. the "outer" border is fixed as white).
  • Puzzle Harder: Match, and also have just a single border. (I.e. neglecting corners and center, there is only one black and one white blob.)
  • My question: I already tried 50 years ago but never verified or even gave a proof. What is the maximum and minimum black area of a valid solution (also counting the center and corners)? You can easily give an upper bound by adding all piece areas, but the value is not obtainable due to parity problems.
  • $\begingroup$ Following your connection with Loops-like puzzles, these are exactly the tiles where the black-white boundary has no self-intersections. Since black and white are perfectly interchangeable, the real problem is to minimize (resp. maximize) the area enclosed by the loop. $\endgroup$ Sep 21 '21 at 14:44

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