6
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Follow-on from Tiling a rectangle with just the Y pentomino

Two questions:

  • Find the smallest rectangle that can be tiled with an odd number of Y pentominoes, or prove it impossible

  • Find the smallest rectangle that can be tiled with an odd number of just 'right-handed' Y pentominoes, i.e. no 'flipping', or prove it impossible

Here is a 5x10 tiled with right-handed Y pentominoes, by way of illustration. All that prevents it from being a valid answer to both questions, is the fact that there is an even number of them.

Y5_10_noflip

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  • $\begingroup$ Need the solutions be proven smallest? This sounds difficult with no-computers. $\endgroup$ – noedne May 23 '18 at 21:48
  • $\begingroup$ @noedne smallest proof not required, smallest will probably be easiest anyway... $\endgroup$ – theonetruepath May 24 '18 at 0:33
  • $\begingroup$ One observation for the 2nd question: both sides of the rectangle must be odd (obviously) but also divisible by 5 (otherwise, gaps will emerge close to the border). $\endgroup$ – Glorfindel May 25 '18 at 10:58
  • $\begingroup$ I've stopped searching for a solution to part 2. In fact I believe I have a fairly simple parity/square-numbering impossibility proof if anyone wants to have a go at getting that. I'll post it in a day or two. $\endgroup$ – theonetruepath May 30 '18 at 8:55
  • $\begingroup$ ...It still eludes me... not as close as I thought. $\endgroup$ – theonetruepath Jun 1 '18 at 0:30
5
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This is not really an answer [technically it is] ... but the answer to Q1 can be found here:

15x15 solution

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  • $\begingroup$ Does finding the answer online count as using no-computers? $\endgroup$ – noedne May 23 '18 at 23:04
  • $\begingroup$ do i have remove this answer ? $\endgroup$ – Q̞ī̯X̶͇͇͇͇͇͇͇͇͇͇͇͇͇̯̯̳̳͈͈͈͆͆ May 23 '18 at 23:06
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    $\begingroup$ I don't know why you say this is "not really an answer". It looks like a valid answer to me, albeit a partial one. $\endgroup$ – Rand al'Thor May 23 '18 at 23:07
  • $\begingroup$ @Q̞ī̯X̶͇͇͇͇͇͇͇͇͇͇͇͇͇̯̯̳̳͈͈͈͆͆ No, I really don't know whether it does, and was hoping someone else could answer. $\endgroup$ – noedne May 23 '18 at 23:10
  • $\begingroup$ Not really an answer mainly because it uses both right- and left-handed Y pentominoes, as is easily seen by examining the three Ys with a long side on the top... they go right, left, right $\endgroup$ – theonetruepath May 24 '18 at 0:37

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