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I'm wondering if it is possible to make a 3x8 grid using 6 tetrominoes: 2 Is, 2 Ls, a Z skew, and a square tetromino. I believe it may be impossible but would like to know why.

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  • $\begingroup$ Even if the L tetronimoes are both right Ls and not a left and right L? I can't see how without flipping the tetronimoes $\endgroup$ Dec 6, 2023 at 19:58
  • $\begingroup$ Can you please give an explicit definition of the tiles, or if those are standard, a link to a description? $\endgroup$
    – Plop
    Dec 7, 2023 at 8:52
  • $\begingroup$ @Plop - They are standard names, I've added a link to the wikipedia page $\endgroup$
    – fljx
    Dec 7, 2023 at 9:05

3 Answers 3

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It is

possible:

    1 1 2 2 2 2 3 3
    4 1 1 5 5 5 3 3
    4 4 4 5 6 6 6 6
 

Without flipping an "L":

    1 1 1 1 2 2 3 3
    4 4 5 5 5 2 2 3
    4 4 5 6 6 6 6 3
 

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  • $\begingroup$ Thank you. I can't seem to make that one myself I think my L shapes are orientated differently but using your answer I have made the following $\endgroup$ Dec 6, 2023 at 19:50
  • $\begingroup$ Problem I have is my L tetronimoes are both of the form of 5 and not 4 so I can't create a grid $\endgroup$ Dec 6, 2023 at 19:53
  • $\begingroup$ @JohnWilliams That's possible, too, see updated answer. $\endgroup$ Dec 6, 2023 at 20:05
  • $\begingroup$ Fantastic. Is this unique or is there another that could be formed? $\endgroup$ Dec 6, 2023 at 20:10
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All the ways it can be tiled. Five ways, if you want to search by hand... Two ways use a mirrored pair of Ls. I have a program that does this sort of thing, started writing it when the IBM PC was released so it's out of control now in terms of code length/maintainability.

enter image description here

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  • $\begingroup$ You should note that only one of those uses two Ls and a Z. The flipped versions of these are J and S. $\endgroup$ Dec 7, 2023 at 13:18
  • $\begingroup$ @DanielMathias Since OP referred to 'tetrominoes' not 'one-sided tetrominoes', I take 'L' to mean 'L or it's mirror'. This is the generally accepted terminology. Of course I accept that OP might have intended it the way you said too... $\endgroup$ Dec 7, 2023 at 17:29
  • $\begingroup$ OP's comments indicate one-sided tetrominoes. That's why I suggested the note. $\endgroup$ Dec 7, 2023 at 17:35
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It is possible; I wrote a small C++ program to find a solution: https://godbolt.org/z/rdjb7n66s

Sample solution:

 L I I I I L O O
 L Z Z L L L O O
 L L Z Z I I I I
 

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