Tiling a rectangle with an odd number of Y pentomoes

Question

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.

Need the solutions be proven smallest? This sounds difficult with [tag:no-computers]. — noedne, May 23 '18 at 21:48
@noedne smallest proof not required, smallest will probably be easiest anyway... — theonetruepath, May 24 '18 at 00:33
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). — Glorfindel, May 25 '18 at 10:58
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. — theonetruepath, May 30 '18 at 08:55

score 5 · Accepted Answer · edited May 24 '18 at 06:23

5

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

edited May 24 '18 at 06:23

Rubio

41,676
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answered May 23 '18 at 23:02

Q̞ī̯X̶͇͇͇͇͇͇͇͇͇͇͇͇͇̯̯̳̳͈͈͈͆͆

401
3
11

Does finding the answer online count as using [tag:no-computers]? – noedne May 23 '18 at 23:04
do i have remove this answer ? – Q̞ī̯X̶͇͇͇͇͇͇͇͇͇͇͇͇͇̯̯̳̳͈͈͈͆͆ May 23 '18 at 23:06
1

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. – Rand al'Thor May 23 '18 at 23:07
@Q̞ī̯X̶͇͇͇͇͇͇͇͇͇͇͇͇͇̯̯̳̳͈͈͈͆͆ No, I really don't know whether it does, and was hoping someone else could answer. – noedne May 23 '18 at 23:10
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 – theonetruepath May 24 '18 at 00:37
Actually i wrote "not really an answer" because the answer not originally solve by me ,but now it make me confiuse what the questions really are. – Q̞ī̯X̶͇͇͇͇͇͇͇͇͇͇͇͇͇̯̯̳̳͈͈͈͆͆ May 24 '18 at 06:03
I was confussed... this is an answer to the first question, my bad. – theonetruepath May 24 '18 at 06:12
I think this is a valid answer. [no-computers] is meant to discourage programming to solve the question (see also the tag wiki). This answer would be equally valid if you looked it up in a book. – Glorfindel May 24 '18 at 08:50
BTW, my computer search confirmed this to be the optimal solution to the first part of the question. – Glorfindel May 24 '18 at 09:49
The second part is quite a bit trickier... – theonetruepath May 24 '18 at 09:58
I know. I suspect it's impossible but I lack the time to prove it mathematically. – Glorfindel May 24 '18 at 11:01

Tiling a rectangle with an odd number of Y pentomoes

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