Thursday, November 3, 2016

geometry - Tiling rectangles with V pentomino plus rectangles


Inspired by Polyomino Z pentomino and rectangle packing into rectangle


Also in this series: Tiling rectangles with F pentomino plus rectangles


Tiling rectangles with N pentomino plus rectangles


Tiling rectangles with T pentomino plus rectangles


Tiling rectangles with U pentomino plus rectangles



Tiling rectangles with W pentomino plus rectangles


Tiling rectangles with X pentomino plus rectangles


The goal is to tile rectangles as small as possible with the V pentomino. Of course this is impossible, so we allow the addition of copies of a rectangle. For each rectangle $a\times b$, find the smallest area larger rectangle that copies of $a\times b$ plus at least one V-pentomino will tile. Examples shown, with the $1\times 1$, $1\times 2$ or $2\times 2$, you can tile a $3\times 3$ as follows:


V plus 1x1, 1x2, 1x3


Now we don't need to consider $1\times 1$, $1\times 2$, or $2\times 2$ any longer as we have found the smallest rectangle tilable with copies of V plus copies of each of those three.


There are at least 20 more solutions. I tagged it 'computer-puzzle' but you can certainly work some of these out by hand. The larger ones might be a bit challenging.



Answer



Here are (most of) the remaining ones. An easy one for 1x5:



enter image description here




and a more difficult one for 1x6:



enter image description here



1x7 takes a lot more:



24x11 = 264

enter image description here



1x8:




16x11 = 176

enter image description here



1x9:



22x12 = 264
enter image description here



1x10:




13x30 = 390

enter image description here



1x12:



14x42 = 588

enter image description here



2x3:



4x8 = 32

enter image description here




2x5:



5x6 = 30

enter image description here



2x6:



8x12 = 96

enter image description here



2x7:




16x19 = 304

enter image description here



2x8:



17x18 = 306

enter image description here



2x9:



21x24 = 504

enter image description here




3x4:



7x18 = 126
enter image description here



3x5:



11x35 = 385

enter image description here



3x7:




13x48 = 624

enter image description here



3x8:



33x38 = 1254

enter image description here



4x5:



21x40 = 840

enter image description here




4x6:



26x36 = 936
enter image description here



5x6:



38x60 = 2280

enter image description here




I assume the number of solutions here is infinite (probably in both directions), I'll post more when I have them.


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