6
$\begingroup$

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

Improve this question
$\endgroup$
5
  • $\begingroup$ 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
$\begingroup$

This is not really an answer [technically it is] ... but the answer to Q1 can be found here:

15x15 solution

Improve this answer
$\endgroup$
11
  • $\begingroup$ 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
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.