Ed’s nontiling set

In[]:=
tiles32=Partition[#,3]&/@Table[IntegerDigits[n,2,6],{n,1,62}];​​longset={1,2,3,4,5,6,7,8,10,12,13,14,16,18,20,22,24,25,26,27,28,29,30,31,34,35,36,37,38,39,42,43,44,46,47,48,50,52,54,57,58,59,60,62};
In[]:=
TilingPatternPlot@tiles32[[longset]]
Out[]=
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In[]:=
Table[{2^n,Timing[GenerateTiling[tiles32[[longset]],{},2^n]]},{n,0,6}]
Out[]=
In[]:=
{#[[1]],#[[2,1]]}&/@%
Out[]=
{{1,0.000165},{2,0.00008},{4,0.026241},{8,0.111218},{16,0.365356},{32,1.46835},{64,6.06629}}
In[]:=
ListLinePlot[%31,PlotRangeAll]
Out[]=
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In[]:=
GenerateTiling[tiles32[[longset]],{},5]
Out[]=
{{1,0,0,1,1},{1,0,0,0,0},{1,0,1,1,1},{1,0,0,1,1},{1,0,0,0,0}}
In[]:=
GenerateTiling[tiles32[[longset]],{},4]
Out[]=
{{0,0,0,0},{1,1,1,1},{0,0,1,1},{0,0,0,0}}
In[]:=
TilingStatePlot[GenerateTilingSeq[{{{0,0,1},{_,1,_}},{{0,1,0},{_,1,_}},{{0,1,1},{_,0,_}},{{1,0,0},{_,0,_}},{{1,0,1},{_,1,_}},{{1,1,0},{_,0,_}}},{},20,3]]
Out[]=
In[]:=
Timing[GenerateTilingSeq[{{{0,0,1},{_,1,_}},{{0,1,0},{_,1,_}},{{0,1,1},{_,0,_}},{{1,0,0},{_,0,_}},{{1,0,1},{_,1,_}},{{1,1,0},{_,0,_}}},{},20,3]]
Out[]=
{0.100809,{{0,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,1,1,0},{0,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0,1},{1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,1,1,0,1,1},{0,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0,1,0,0},{0,1,0,0,1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1},{1,1,0,1,0,0,1,1,0,1,0,0,1,0,0,1,0,0,1,0},{0,0,1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,1,0},{0,1,0,0,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1},{1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1},{0,0,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0},{0,1,0,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1},{1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,1},{0,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0},{0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,1,0,1},{1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,1,1},{0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0},{0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,1,1,0,1},{1,0,0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0,1,1},{1,0,1,1,0,1,1,0,1,1,0,1,0,0,1,1,0,1,0,0},{0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0,1,1,0,1}}}

Tiling tests with sequential checks

In[]:=
FindTilingsSeq[({{#1,#2,_},{#3,_,#4},{#5,_,_}}&@@@Tuples[{1,0},5]),32,20,{}]
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3.65324
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0.01085
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435
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Try with parallelism enabled for NewTilings0:
(The number of intermediate candidates seems to be of order Binomial[32, n]/2.)

Reverse Engineering from Graphics