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You are given 7 identical copies of the following polyomino (made of unit squares):

![polyomino](https://github.com/kateyose/Tilted-Packing-Puzzle/blob/main/misc/piece2.svg)

Goal is to place all 7 pieces inside a square tray of side length: 8.5 units

  • You may rotate by any angle and flip pieces.
  • Overlaps are not allowed.
  • Gaps are allowed (it does not have to be a perfect tiling).

Question: Can you find a placement of the 7 pieces that fits inside the square?

Disclosure: Design notes are on GitHub (not needed to solve):

https://github.com/kateyose/Tilted-Packing-Puzzle

If you liked this, other puzzles in the same series: Puzzle #1, #3, and #4: (link)

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    $\begingroup$ Welcome to Puzzling, and nice first puzzle! As your link contains solutions, you might want to omit it until someone solves the puzzle. $\endgroup$ Commented yesterday
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    $\begingroup$ I have something close but not quite there yet: i.sstatic.net/pBGYPicf.png $\endgroup$ Commented 21 hours ago

1 Answer 1

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8.5 just screams to be multiplied by √5, and it is just over 19/√5.

enter image description here

Here's is Pranay's attempt: enter image description here

With some slight changes we get: enter image description here

which works.

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  • $\begingroup$ Congratulations — yes, that's the intended solution. For a computer solver, two approaches: (1) enumerate feasible tray grid patterns first (as you did, but there are more patterns actually), then search; (2) a more random/continuous method exploiting the polyomino being star-shaped (a billiard-style algorithm). $\endgroup$ Commented 14 hours ago
  • $\begingroup$ If you liked this, other puzzles in the same series on my GitHub: github.com/kateyose/Tilted-Packing-Puzzle $\endgroup$ Commented 14 hours ago
  • $\begingroup$ Ref.1 (Japanese): hyoutan-daisuki.hatenablog.com/entry/2024/08/19/171731 Ref.2 : Improved Dense Packings of Congruent Squares in a Square: link.springer.com/article/10.1007/s00454-004-1129-z $\endgroup$ Commented 14 hours ago
  • $\begingroup$ +1. Wonderful! I’m glad that I at least got the tilt right! $\endgroup$ Commented 14 hours ago

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