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Unpainting Black (Posted on 2017-07-14) Difficulty: 4 of 5
Like in Paint it Black, twenty-seven identical white cubes are assembled into a single cube; then the outside of that cube is painted black.

The cube is then disassembled and rebuilt randomly.

What is the probability that the outside of this cube is now completely white?

No Solution Yet Submitted by Brian Smith    
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Details of the results Comment 5 of 5 |
In order to assist in checking on the validity of the results of the program without going through the program logic, I present below the intermediate result "matrices" indicating the probability of getting to a given state, i.e. distribution of remaining cubelets, after each of the stages.

After the first stage, selecting the new corner pieces, there are 77 possible  distributions of the remaining 19 cubelets with regard to original location, that is after a successful completion of phase 1, i.e., no black cubelet faces showing on the outer cube surface.  After the second stage, selecting the new edge pieces, there are 62 possible (after successful phases 1 and 2) remaining distributions. And of course on completion there are only 4 possible identities of the remaining cube to be in the center.

This also shows there were only computations to be made, altogether, for 1+77+62=140 possible sets of available distributions.

Incidentally also, the last set shows the probabilities you'll wind up with a given type of cubelet in the center when the task is successfully completed. (overall probability that is, not conditional). These probabilities are what add up to the answer (assuming no errors).

Each row below identifies the stage by the number of remaining cubelets, then how many of each type: corner, edge, face and center, and finally the probability of having successfully reached that state. The E should be interpreted as *10^.

19  0 12 6 1   6.65873811009515E-19 
19  1 11 6 1   1.02278217371062E-15 
19  1 12 5 1   1.02278217371062E-15 
19  1 12 6 0   6.81854782473744E-15 
19  2 10 6 1   2.75639795815011E-13 
19  2 11 5 1   6.01395918141842E-13 
19  2 11 6 0   4.00930612094561E-12 
19  2 12 4 1   5.01163265118202E-13 
19  2 12 5 0   4.00930612094561E-12 
19  3 9 6 1   2.20511836652009E-11 
19  3 10 5 1   7.93842611947231E-11 
19  3 10 6 0   5.29228407964821E-10 
19  3 11 4 1   1.44335020354042E-10 
19  3 11 5 0   1.15468016283234E-09 
19  3 12 3 1   9.62233469026947E-11 
19  3 12 4 0   9.62233469026947E-10 
19  4 8 6 1   6.20189540583775E-10 
19  4 9 5 1   3.30767754978013E-09 
19  4 9 6 0   2.20511836652009E-08 
19  4 10 4 1   9.92303264934039E-09 
19  4 10 5 0   7.93842611947232E-08 
19  4 11 3 1   1.44335020354042E-08 
19  4 11 4 0   1.44335020354042E-07 
19  4 12 2 1   7.21675101770211E-09 
19  4 12 3 0   9.62233469026947E-08 
19  5 7 6 1   6.35074089557785E-09 
19  5 8 5 1   4.76305567168339E-08 
19  5 8 6 0   3.17537044778893E-07 
19  5 9 4 1   2.11691363185928E-07 
19  5 9 5 0   1.69353090548743E-06 
19  5 10 3 1   5.08059271646228E-07 
19  5 10 4 0   5.08059271646228E-06 
19  5 11 2 1   5.54246478159522E-07 
19  5 11 3 0   7.38995304212695E-06 
19  5 12 1 1   1.84748826053174E-07 
19  5 12 2 0   3.69497652106348E-06 
19  6 6 6 1   2.22275931345225E-08 
19  6 7 5 1   2.28626672240803E-07 
19  6 7 6 0   1.52417781493868E-06 
19  6 8 4 1   1.42891670150502E-06 
19  6 8 5 0   1.14313336120401E-05 
19  6 9 3 1   5.08059271646228E-06 
19  6 9 4 0   5.08059271646228E-05 
19  6 10 2 1   9.14506688963211E-06 
19  6 10 3 0   1.21934225195095E-04 
19  6 11 1 1   6.65095773791426E-06 
19  6 11 2 0   1.33019154758285E-04 
19  6 12 0 1   1.10849295631904E-06 
19  6 12 1 0   4.43397182527617E-05 
19  7 5 6 1   2.17739687848384E-08 
19  7 6 5 1   3.04835562987737E-07 
19  7 6 6 0   2.03223708658491E-06 
19  7 7 4 1   2.6128762541806E-06 
19  7 7 5 0   2.09030100334448E-05 
19  7 8 3 1   1.3064381270903E-05 
19  7 8 4 0   1.3064381270903E-04 
19  7 9 2 1   3.48383500557414E-05 
19  7 9 3 0   4.64511334076551E-04 
19  7 10 1 1   4.18060200668896E-05 
19  7 10 2 0   8.36120401337793E-04 
19  7 11 0 1   1.52021891152326E-05 
19  7 11 1 0   6.08087564609304E-04 
19  7 12 0 0   1.01347927434884E-04 
19  8 4 6 1   3.40218262263099E-09 
19  8 5 5 1   6.5321906354515E-08 
19  8 5 6 0   4.35479375696767E-07 
19  8 6 4 1   7.62088907469342E-07 
19  8 6 5 0   6.09671125975474E-06 
19  8 7 3 1   5.2257525083612E-06 
19  8 7 4 0   5.2257525083612E-05 
19  8 8 2 1   1.95965719063545E-05 
19  8 8 3 0   2.6128762541806E-04 
19  8 9 1 1   3.48383500557414E-05 
19  8 9 2 0   6.96767001114827E-04 
19  8 10 0 1   2.09030100334448E-05 
19  8 10 1 0   8.36120401337793E-04 
19  8 11 0 0   3.04043782304652E-04 

7  0 0 6 1   5.36417137534667E-21 
7  0 1 5 1   6.77886265983132E-19 
7  0 1 6 0   2.7542260813075E-18 
7  0 2 4 1   2.40915186846615E-17 
7  0 2 5 0   1.58719323683968E-16 
7  0 3 3 1   3.62729266379587E-16 
7  0 3 4 0   3.41689578134974E-15 
7  0 4 2 1   2.55444729744794E-15 
7  0 4 3 0   3.4911899224931E-14 
7  0 5 1 1   8.09417745237993E-15 
7  0 5 2 0   1.76837768247788E-13 
7  0 6 0 1   9.01392674646833E-15 
7  0 6 1 0   4.15141237339177E-13 
7  0 7 0 0   3.45591199462209E-13 
7  1 0 5 1   3.46859568737771E-18 
7  1 0 6 0   1.41292060352391E-17 
7  1 1 4 1   2.69191928074336E-16 
7  1 1 5 0   1.77769241647104E-15 
7  1 2 3 1   6.69506599127023E-15 
7  1 2 4 0   6.3188904060906E-14 
7  1 3 2 1   6.99334907001164E-14 
7  1 3 3 0   9.56938765165376E-13 
7  1 4 1 1   3.11976287275064E-13 
7  1 4 2 0   6.81721226866216E-12 
7  1 5 0 1   4.77022410388329E-13 
7  1 5 1 0   2.19472204380141E-11 
7  1 6 0 0   2.48734184213048E-11 
7  2 0 4 1   5.32909011255444E-16 
7  2 0 5 0   3.52698370032368E-15 
7  2 1 3 1   2.89495291213649E-14 
7  2 1 4 0   2.73716059250736E-13 
7  2 2 2 1   4.99542431231586E-13 
7  2 2 3 0   6.84311333616245E-12 
7  2 3 1 1   3.30526911101275E-12 
7  2 3 2 0   7.22397597400711E-11 
7  2 4 0 1   7.11510660626931E-12 
7  2 4 1 0   3.27066380440404E-10 
7  2 5 0 0   5.08472660985452E-10 
7  3 0 3 1   2.84701040833926E-14 
7  3 0 4 0   2.69628973173549E-13 
7  3 1 2 1   1.07312309050781E-12 
7  3 1 3 0   1.47156748692268E-11 
7  3 2 1 1   1.17292709104759E-11 
7  3 2 2 0   2.56405096599784E-10 
7  3 3 0 1   3.74481929643239E-11 
7  3 3 1 0   1.72006952408405E-09 
7  3 4 0 0   3.76142407743335E-09 
7  4 0 2 1   5.60582438382749E-13 
7  4 0 3 0   7.69474683743274E-12 
7  4 1 1 1   1.33838761268962E-11 
7  4 1 2 0   2.92636202948699E-10 
7  4 2 0 1   7.05845515182259E-11 
7  4 2 1 0   3.23986905471281E-09 
7  4 3 0 0   1.04991134537014E-08 
7  5 0 1 1   3.68690808019055E-12 
7  5 0 2 0   8.06333591294031E-11 
7  5 1 0 1   4.24706257104666E-11 
7  5 1 1 0   1.94829194067475E-09 
7  5 2 0 0   1.042065007272E-08 
7  6 0 0 1   5.7675282834593E-12 
7  6 0 1 0   2.64449668756428E-10 
7  6 1 0 0   3.0871633047416E-09 
7  7 0 0 0   1.77998113458635E-10 

1  0 0 0 1   1.2312327798997E-13 
1  0 0 1 0   5.61961327833826E-12 
1  0 1 0 0   5.06496356523964E-11 
1  1 0 0 0   1.67317232060851E-10 


  Posted by Charlie on 2017-07-15 11:29:16
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