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Problem 671

Posted: Tue May 21, 2019 1:36 pm
by amagri
Hello,

On the last problem, I wanted some clarifications on the first example F4(3)=104.

Question 1:
Is it authorized to have a tile (eg. 1x3) adjacent to itself?
« Adjacent tiles must be of different colours », but here we only have 1 tile.

Question 2:
I have a similar question regarding the 2nd rule: « It is not permitted for four tiles to have their corners meeting at a single point ». For n=3, we may have « four corners meeting at a point », but with only 2 or 3 tiles. Is it also forbidden?

Many thanks in advance

PS: these clarifications only concern cases where n<=3

Re: Problem 671

Posted: Mon May 27, 2019 11:33 am
by MuthuVeerappanR
Problem 671 (View Problem)

Hi All,
Tiling which are identical after rotating the loop about its axis considered different? Only if I consider them different, I get F_4(3) = 104. Else am stuck at 92.

I'm not sure whether 'reflecting horizontally or vertically would give a different tiling, these tilings are to be counted separately' covers this or not.

Can someone please clarify?

Thanks
MuthuVeerappanR

Re: Problem 671

Posted: Mon May 27, 2019 1:40 pm
by jaap
I have not solved this question yet, but have worked out F_4(3) on paper.

@amagri:
I believe those are indeed forbidden, so for both those reasons you cannot use a 1x3 tile when n=3.

@MuthuVeerappanR:
I think rotations around the axis of the loop are not considered different. I get 24+24+24+24+8 = 104 tilings for this. If rotations were different, I'd get 72+72+72+72+24 = 312 tilings.

In my opinion the question should also state that turning the loop upside down, just like horizontal or vertical reflection (and is in fact equivalent to doing both reflections), is also considered to be different. If it were not, I think that for F_4(3) there would be only 104/2 = 52 tilings.

Re: Problem 671

Posted: Tue May 28, 2019 8:47 am
by amagri
Thank you jaap for your inputs.

I have not yet solved problem #671, but I am now able to compute F4(3), as well as the other provided examples.

I confirm the points:
- when rotated with rotations Ri (i=1..n-1), loops don’t lead to different loops
- all other transformations (vertical or horizontal reflections, rotating/turning the loop upside down) lead to different loops
- for small n, a tile cannot be adjacent to itself, and the "no four corners meet at a point" rule is applicable regardless of the number of tiles involved

Thank you to both of you for this exchange, and mutual support, good luck!