The New Sudoku Players' Forum

[Mathimagics]

An intercalate in a Latin Square is a 2x2 sub-square that contains just 2 values, that is (rx, cx) = (ry, cy) and (rx, cy) = (ry, cx). I will call these "LIC"s here.
If we regard a given Sudoku grid as a Latin Square, all UA4's on the grid are obviously LIC's. But the converse is not always true. Our Sudoku grid might have an LIC on the corner cells, (r1,c1) = (r9,c9) and (r1,c9) = (r9,c1), but that can't be a UA4 as the cells are in 4 different boxes.

Of course, the idea of a UA is applicable to Latin Squares, and all LIC's on a Sudoku grid are unavoidable sets (of size 4) on the Latin Square, but the term "unavoidable" appears to mean something different in the published literature. For example, in this short paper, which is a gentle introduction to critical sets (which we could call minimal puzzles), the concept of UA's is implied, but is not given a name.

Anyway, my interest was sparked here by a reference in the LCT project thread to this example of a grid with 36 UA4's:

Code: Select all

123568479864791352957243681218657934536489127749312865391825746472136598685974213

It has some interesting properties. The UA4's cover all 81 cells, the grid has 36 automorphisms, and MCN = 15.

It also has exactly 36 intercalates.

I thought that this was worth some further investigation, and began by counting the LIC's and UA4's for every ED grid in the catalog. I also separately counted them for the 560,151 NTA grids (grids with non-trivial automorphisms). Finally I added the LIC counts for a large sample of random Latin Squares.

This confirmed that:

• 36 UA4's is absolutely the maximum possible

• there are 9 ED grids that have 36 UA4's

It also turned up other interesting items, which I will come to in due course. The full table of results is given below, it shows for each N = 0, 1, 2 ... the number of grids in the set that were found to have exactly N LIC's and UA4's.

A summary:

Code: Select all

GPLS: (gen pop = random Latin squares)
  NG:   5474615294  (sample)
  Avg:     17.8755 LIC's / grid
  Max # of LIC = 62 (observed)

ED: Sudoku catalog
  NG:   5472730538
  Avg:     19.6259 LIC's / grid
           11.5788 UA4's / grid
  Max # of LIC = 72, max # of UA4 = 36

NTA: (automorphic grids)
  NG:       560151
  Avg:     18.6205 LIC's / grid
            9.5769 UA4's / grid
  Max # of LIC = 72, max # of UA4 = 36


LIC v UA4 distribution: Show

Code: Select all

 +----+------------+--------------+--------------+------------+------------+-----------+-----------+
 |    |  LS  grids |    ED grids  |  ED  grids   |  NTA grids | NTA  grids | NTA  odd  | NTA odd   |
 | N  |  N x LIC's |   N x LIC's  |  N x UA4's   |  N x LIC's | N  x UA4's | N x LIC's | N x UA4's |
 +----+------------+--------------+--------------+------------+------------+-----------+-----------+
 |    |       445  |        3484  |     352894   |       1789 |      18671 |           |           |
 |  1 |      7004  |        6385  |    1654475   |            |            |        0  |        0  |
 |  2 |     55305  |       35653  |    6891532   |       1869 |      23133 |           |           |
 |  3 |    283129  |      141507  |   21107315   |            |            |      269  |     3150  |
 |  4 |   1102732  |      512521  |   51765007   |       4847 |      48584 |           |           |
 |  5 |   3429506  |     1552981  |  105759561   |            |            |      747  |     4554  |
 |  6 |   8982935  |     4105054  |  186644998   |      10859 |      71825 |           |           |
 |  7 |  20261617  |     9452319  |  289799298   |            |            |      908  |     5630  |
 |  8 |  40331452  |    19553037  |  402555715   |      19166 |      85015 |           |           |
 |  9 |  71899242  |    36451912  |  505627121   |            |            |     2551  |     6597  |
 | 10 | 116407345  |    62285461  |  580153348   |      28807 |      85680 |           |           |
 | 11 | 172949562  |    98137605  |  612280600   |            |            |     2595  |     4848  |
 | 12 | 237901341  |   143970856  |  598177291   |      42741 |      72679 |           |           |
 | 13 | 305297079  |   197651898  |  543232929   |            |            |     4406  |     3605  |
 | 14 | 367903539  |   255803641  |  460415833   |      51038 |      51738 |           |           |
 | 15 | 418525183  |   313054783  |  364777309   |            |            |     6155  |     2926  |
 | 16 | 451642514  |   364242776  |  270755777   |      56931 |      32544 |           |           |
 | 17 | 464238679  |   404037147  |  188260086   |            |            |     5426  |     1145  |
 | 18 | 456105034  |   428953009  |  122672303   |      61079 |      19393 |           |           |
 | 19 | 429667681  |   436712304  |   74788269   |            |            |     5240  |      514  |
 | 20 | 389086590  |   427716042  |   42628010   |      55765 |       9523 |           |           |
 | 21 | 339509998  |   403380517  |   22634443   |            |            |     5796  |      412  |
 | 22 | 286063286  |   367343148  |   11177578   |      47509 |       4495 |           |           |
 | 23 | 233125278  |   323178833  |    5123654   |            |            |     3999  |       60  |
 | 24 | 184014869  |   275344231  |    2176664   |      39833 |       2178 |           |           |
 | 25 | 140924022  |   227169934  |     854253   |            |            |     3231  |       10  |
 | 26 | 104805024  |   181868288  |     311585   |      29813 |        790 |           |           |
 | 27 |  75715049  |   141217990  |     105410   |            |            |     2932  |       28  |
 | 28 |  53250289  |   106552647  |      33612   |      20737 |        281 |           |           |
 | 29 |  36432047  |    78076764  |       9897   |            |            |     1703  |           |
 | 30 |  24277171  |    55666966  |       2798   |      15308 |        103 |           |           |
 | 31 |  15760808  |    38533843  |        721   |            |            |      995  |           |
 | 32 |   9973086  |    25982625  |        186   |       9438 |         25 |           |           |
 | 33 |   6141453  |    17008465  |         43   |            |            |      759  |        3  |
 | 34 |   3692062  |    10859297  |         14   |       5738 |          3 |           |           |
 | 35 |   2160291  |     6728751  |              |            |            |      256  |           |
 | 36 |   1236025  |     4069483  |          9   |       3862 |          9 |           |           |
 | 37 |    684484  |     2382599  |              |            |            |      210  |           |
 | 38 |    373867  |     1370243  |              |       2112 |            |           |           |
 | 39 |    196159  |      756831  |              |            |            |      145  |           |
 | 40 |    102268  |      416356  |              |       1106 |            |           |           |
 | 41 |     51190  |      216407  |              |            |            |       28  |           |
 | 42 |     25541  |      113900  |              |        660 |            |           |           |
 | 43 |     12141  |       56730  |              |            |            |        5  |           |
 | 44 |      5934  |       28529  |              |        364 |            |           |           |
 | 45 |      2688  |       13334  |              |            |            |       14  |           |
 | 46 |      1315  |        6868  |              |        182 |            |           |           |
 | 47 |       554  |        3180  |              |            |            |           |           |
 | 48 |       269  |        1652  |              |        119 |            |           |           |
 | 49 |       118  |         873  |              |            |            |           |           |
 | 50 |        52  |         431  |              |         27 |            |           |           |
 | 51 |        17  |         156  |              |            |            |        3  |           |
 | 52 |        12  |         148  |              |         36 |            |           |           |
 | 53 |         7  |          45  |              |            |            |           |           |
 | 54 |         3  |          51  |              |         19 |            |           |           |
 | 55 |         0  |           0  |              |            |            |           |           |
 | 56 |            |          23  |              |          8 |            |           |           |
 | 57 |            |           5  |              |            |            |           |           |
 | 58 |         2  |          16  |              |         13 |            |           |           |
 +----+------------+--------------+--------------+------------+------------+-----------+-----------+
 | 62 |         1  |           1  |              |            |            |           |           |
 | 66 |            |           2  |              |          2 |            |           |           |
 | 72 |            |           1  |              |          1 |            |           |           |
 +----+------------+--------------+--------------+------------+------------+-----------+-----------+


[Mathimagics]

There are many intriguing points of interest in this table. I made the sample size of Latin Squares the same as our Sudoku ED catalog so the LIC frequencies would be comparable. But the ED set are just that, they are ED while the Latin Squares are not, and I have no way of determining how many ED LS's that the sample contains.

So, while the table might suggest that LIC-free grids occur more frequently for Sudoku grids than in the general population of Latin Squares, this is probably an illusion.

The normal distribution shape is evident in each set, but for the NTA grids this only becomes clear if you treat even/odd N cases separately, which is why I listed them that way. And the NTA set has a blank spot for N = 1. This is the only "hole" in the table until we hit N = 55 (which itself is of interest because N = 55 has no entries for any set).

Here are the 9 grids with 36 UA4's. I have included counts for # of LIC's, # of automorphisms, and # of cells covered by the UA4's (CC).

Code: Select all

                                                                                   LIC UA4   NA  CC
123456789457189236698372514215768943389541672764293851572634198831927465946815327 # 38  36    2  75
123456789457189236698372514269541873314798652785263941541627398876934125932815467 # 38  36    2  75
123456789457189236869327415296743851378512964541698327682971543734265198915834672 # 52  36    4  69
123456789457189236968372514215694378386527491794813625572941863649738152831265947 # 36  36    6  75
123456789457189236968372514215734968386921457794865123572698341649213875831547692 # 54  36   18  72
123456789457189236968372514219547368586213497734698125372865941645921873891734652 # 66  36   12  75
123456789457189236968372514219634857586927143734815962372541698645798321891263475 # 36  36    6  75
123456789457189326869372514214965873635847192978213465381624957546798231792531648 # 36  36   36  81* orig example
123456789457189236968372514291738465374265198685941327546813972732694851819527643 # 72  36   36  81*


The last grid has 72 LIC's, which makes it far and away the highest LIC count seen in any Latin square, and of course it is the only such case among the Sudoku grids. It turns out to have MCN = 12, which was a little bit disappointing ...

[Mathimagics]

Interesting grids #1:

The MC grid has no UA4's, but here is one that is totally LIC-free and which is NOT in the NTA set.

Code: Select all

123456789456789123789123456234891567597264831618375942375942618841637295962518374


• NA = 1 (no NTA), no LIC's (and so no UA4's)
• UAs: 41 ua6, 16 ua8, 7 ua9, 58 ua10, 34 ua11, 161 ua12
• MCN = 9 (1468 ways, 729 using only the ua6's)

Code: Select all

 +-------+-------+-------+
 | 1 2 3 | 4 5 6 | 7 8 9 |
 | 4 5 6 | 7 8 9 | 1 2 3 |
 | 7 8 9 | 1 2 3 | 4 5 6 |
 +-------+-------+-------+
 | 2 3 4 | 8 9 1 | 5 6 7 |
 | 5 9 7 | 2 6 4 | 8 3 1 |
 | 6 1 8 | 3 7 5 | 9 4 2 |
 +-------+-------+-------+
 | 3 7 5 | 9 4 2 | 6 1 8 |
 | 8 4 1 | 6 3 7 | 2 9 5 |
 | 9 6 2 | 5 1 8 | 3 7 4 |
 +-------+-------+-------+


Interesting grids #2

A grid with no UA4's, but many LIC's (39)s is the maximal case of such a grid.

Code: Select all

123456789456789123789123456231597648564831972897264315372915864615348297948672531


• NA = 3, no UA4, 39 IC's
• UAs: 102 ua6, 57 ua12
• MCN = 12 (8305 ways)

Code: Select all

 +-------+-------+-------+
 | 1 2 3 | 4 5 6 | 7 8 9 |
 | 4 5 6 | 7 8 9 | 1 2 3 |
 | 7 8 9 | 1 2 3 | 4 5 6 |
 +-------+-------+-------+
 | 2 3 1 | 5 9 7 | 6 4 8 |
 | 5 6 4 | 8 3 1 | 9 7 2 |
 | 8 9 7 | 2 6 4 | 3 1 5 |
 +-------+-------+-------+
 | 3 7 2 | 9 1 5 | 8 6 4 |
 | 6 1 5 | 3 4 8 | 2 9 7 |
 | 9 4 8 | 6 7 2 | 5 3 1 |
 +-------+-------+-------+


[dobrichev]

Look at Solution Grids in U-Space (Unavoidable Sets).

Also, the Sudoku variants section may be more appropriate for discussing such topics.

[Mathimagics]

Thanks Mladen, for that first link ...

But I have no idea why you think this thread belongs in "Sudoku Variants" ... ????

[dobrichev]

Because "Latin Square Variants" are in other forums.

[Mathimagics]

Ah, but the Sudoku Puzzle is a variant of the Latin Square Completion puzzle, and not the other way around ...

[Mathimagics]

Mladen's link shows he has been down this path long before me, and it is nice that I am able to confirm that his got his UA4 counts were absolutely right!

It is also pleasing to see that the count of 9 grids with 36 UA4's is confirmed.

And I should add that those colour plots of the ED grid distribution in UA4/6 space are really stunning. Great work!

[coloin]

I think its relevant ... and its been a long time ... a very long time

here max no of clues 2

Code: Select all

 Dukuso15 - 123568479864791352957243681218657934536489127749312865391825746472136598685974213

Code: Select all

    {11,12,41,42,}    {26,29,34,39,65,66,73,75,83,84,}    {57,58,97,98,}
    {11,13,71,73,}    {26,27,36,39,42,48,52,57,98,99,}    {64,65,84,85,}
    {11,16,21,26,}    {38,39,42,43,73,74,84,89,92,98,}    {55,57,65,67,}
    {11,19,31,39,}    {25,26,42,47,56,57,63,65,72,73,}    {84,88,94,98,}
    {12,13,41,48,52,58,71,75,83,85,}    {27,29,97,99,}    {34,36,64,66,}
    {12,14,32,34,}    {28,29,41,45,51,58,66,69,75,76,}    {83,87,93,97,}
    {12,18,24,29,33,34,58,59,82,83,}    {41,46,61,66,}    {75,77,95,97,}
    {13,17,23,27,}    {35,36,48,49,71,78,81,85,96,99,}    {52,54,62,64,}
    {13,15,22,27,36,37,52,53,85,86,}    {44,48,64,68,}    {71,79,91,99,}
    {15,17,35,37,}    {22,23,44,49,53,54,62,68,78,79,}    {81,86,91,96,}
    {17,18,77,78,}    {23,24,33,35,46,49,54,59,95,96,}    {61,62,81,82,}
    {17,19,47,49,}    {23,25,31,35,62,63,72,78,81,88,}    {54,56,94,96,}
    {14,15,44,45,}    {22,28,32,37,68,69,76,79,86,87,}    {51,53,91,93,}
    {14,18,24,28,}    {32,33,45,46,76,77,82,87,93,95,}    {51,59,61,69,}
    {14,16,74,76,}    {21,28,32,38,43,45,51,55,92,93,}    {67,69,87,89,}
    {15,16,43,44,53,55,74,79,86,89,}    {21,22,91,92,}    {37,38,67,68,}
    {18,19,46,47,56,59,72,77,82,88,}    {24,25,94,95,}    {31,33,61,63,}
    {16,19,21,25,31,38,55,56,88,89,}    {43,47,63,67,}    {72,74,92,94,}

I looked at the duk15 grid as it had 36UA4s

I too saw the 36 UA and I remove 3 clues from each to see if I could get a 36-clue puzzle .....unfortunately it had 2 solutions and adding a clue made it non minimal....

Code: Select all

+---+---+---+
|...|...|...|
|.64|.91|.52|
|.57|.43|.81|
+---+---+---+
|.18|.57|.34|
|.36|.89|.27|
|...|...|...|
+---+---+---+
|.91|.25|.46|
|.72|.36|.98|
|...|...|...|
+---+---+---+ 36-clue minimal pseudoku with 2 sols

123568479864791352957243681218657934536489127749312865391825746472136598685974213
189572463364891752257643981918257634436189527725364819891725346572436198643918275

Interestingly the other solution grid was a stunner !!!!!!

we went from a grid with 36UAs to one with no U4s and only U6s

Im sure the answer will be blowing in the wind

[dobrichev]

Hi Mathimagics,
Thank you for the good words.

Actually there is unfinished work in these diagrams.
They have few visible gaps in 17/all 2D distribution which can predict the number of unavoidable sets of respective type for the unknown grids having 17-clue puzzles.
Blue found new puzzles after these distributions were calculated, and now it is interesting to see whether their solution grids hit the gaps, this confirming the prediction.
The prediction itself hasn't much practical value, because of the huge number of grids having respective number of UA. But in terms of percentage it is sufficiently restrictive.

[Mathimagics]

we went from a grid with 36UAs to one with no U4s and only U6s

If there are two solutions, then perhaps the fact all the UA4's we started with must overlap to a very great degree, would mean that any solution other than the original must have all of those UA4's disrupted. So solutions are one extreme (many UA4's or the other, none at all).

This is just a thought ... something along those lines ...

[coloin]

Well i just thought it was that a u4 and a u6 cant co-exist in a minirow/minirow ....
I wonder do the other 36U4s partnera grid with all repeating minirow solution grids like the MC grid ?
I will have to get out my old program of dukuso's [unavoid.exe] !

[dobrichev]

... we went from a grid with 36UAs to one with no U4s and only U6s ...

Hi coloin,
I am sure you are the only one who has imagination to do such tricks by hand.

Congratulations!

[coloin]

I am sure you are the only one who has imagination to do such tricks by hand.

confession i do have havards gui program

however these compare the maximum simiaruty of clues with the MC grid

Code: Select all

sudoku-64 -CSf test.txt
similarity 45                                                                                           index416       
189572463364891752257643981918257634436189527725364819891725346572436198643918275 - MC Grid     1   1   1  ,   1   1   1
123568479864791352957243681218657934536489127749312865391825746472136598685974213 - Duk15     271 271 271  , 271 271 271
                                                                                                                       
similarity 41                                                                                                           
189572463364891752257643981918257634436189527725364819891725346572436198643918275 - MC Grid     1   1   1  ,   1   1   1
123594768497681253568273941219857634386149527754362819831726495972435186645918372 - #1          1 320 320  , 354 365 373
similarity 40                                                                                                           
189572463364891752257643981918257634436189527725364819891725346572436198643918275 - MC Grid     1   1   1  ,   1   1   1
489572361365891472127643958812759634943168527576234819298315746751486293634927185 - #2         70 116 116  ,  70 131 131
similarity 41                                                                                                           
189572463364891752257643981918257634436189527725364819891725346572436198643918275 - MC Grid     1   1   1  ,   1   1   1
381792564469581723257643981914237658736859412528164397192475836875326149643918275 - #3        139 139 271  , 139 139 271
similarity 45                                                                                                           
189572463364891752257643981918257634436189527725364819891725346572436198643918275 - MC Grid     1   1   1  ,   1   1   1
617532489482791653395648721168257934234189567759364812821975346576423198943816275 - #4         70  70  70  ,  70  70  70
similarity 54                                                                                                           
189572463364891752257643981918257634436189527725364819891725346572436198643918275 - MC Grid     1   1   1  ,   1   1   1
189472653364895721257613489816257934423189567975364812791528346532746198648931275 - #5         70  70  70  ,  70  70  70
similarity 39                                                                                                           
189572463364891752257643981918257634436189527725364819891725346572436198643918275 - MC Grid     1   1   1  ,   1   1   1
189472365364895172257613948812957634493186527675324819948231756536749281721568493 - #6         70  70  70  ,  70  70  70
similarity 45                                                                                                           
189572463364891752257643981918257634436189527725364819891725346572436198643918275 - MC Grid     1   1   1  ,   1   1   1
489572361365891472217643958958127634136984527724356819842765193671439285593218746 - #7         70  70  70  ,  70  70  70
similarity 45                                                                                                           
189572463364891752257643981918257634436189527725364819891725346572436198643918275 - MC Grid     1   1   1  ,   1   1   1
123568479864791352957243681218657934536489127749312865391825746472136598685974213 - #8 Duk15  271 271 271  , 271 271 271
similarity 45                                                                                                           
189572463364891752257643981918257634436189527725364819891725346572436198643918275 - MC Grid     1   1   1  ,   1   1   1
179568423864231759253749681948657132536182947721394865497825316312476598685913274 - #9         70  70  70  ,  70  70  70


so all these grids share many similar solution grid clues to the MC GRid - and they differ by q 27-clue UA in #5

EDIT
it appears that a random grid can be morphed to on average 38 clues in common with the MC grid

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similarity 38
123456789456789123789123456231564897564897231897231564312645978645978312978312645
329456781156783249487219356531864927264597138798321465812935674643178592975642813

so perhaps all we can say is that those 36U4 grids above have more common clues with the MC grid than a random grid

[coloin]

This grid [blue1] fromgrids containing a 21 but no 20

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123456789456789123789123456214897365365214897897365214541632978632978541978541632  blue1

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+---+---+---+
|123|456|789|
|456|789|123|
|789|123|456|
+---+---+---+
|214|897|365|
|365|214|897|
|897|365|214|
+---+---+---+
|541|632|978|
|632|978|541|
|978|541|632|
+---+---+---+


has these bands

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  1,  1,  1,224,224,224       0,     0,     0,     0,     0,     0

and each band has no way to complete in less than 6 clues [no 5-completions]

it is similar to the mc grid in one direction - full of U6s, but at 90 degrees it is full of U4s [18, 6 in each vertical band] MCN=15

Hidden Text: Show

The maximum # of disjoint unavoidable sets (max clique number -- MCN) is 15.
One such maximal collection is:
{11,12,41,42}
{14,15,94,95}
{21,22,71,72}
{27,28,67,68}
{34,35,54,55}
{37,38,87,88}
{47,48,97,98}
{51,52,81,82}
{64,65,74,75}
{13,16,23,29,36,39}
{17,18,19,57,58,59}
{24,25,26,44,45,46}
{31,32,33,61,62,63}
{43,49,53,56,66,69}
{73,76,79,83,86,89}

some small unavoidables

Hidden Text: Show

{11,12,41,42}
{12,13,82,83}
{14,15,94,95}
{15,16,65,66}
{21,22,71,72}
{22,23,52,53}
{27,28,67,68}
{28,29,98,99}
{34,35,54,55}
{35,36,75,76}
{37,38,87,88}
{38,39,48,49}
{42,43,72,73}
{47,48,97,98}
{51,52,81,82}
{55,56,95,96}
{64,65,74,75}
{68,69,88,89}
{11,13,51,53,71,73}
{11,14,17,21,24,27}
{11,14,17,31,34,37}
{11,14,21,24,31,34}
{11,14,21,27,34,37}
{11,17,21,27,31,37}
{11,17,24,27,31,34}
{12,15,18,22,25,28}
{12,15,18,32,35,38}
{12,15,22,25,32,35}
{12,15,22,28,35,38}
{12,18,22,28,32,38}
{12,18,25,28,32,35}
{13,16,19,23,26,29}
{13,16,19,33,36,39}
{13,16,23,26,33,36}
{13,16,23,29,36,39}
{13,19,23,29,33,39}
{13,19,26,29,33,36}
{14,16,54,56,74,76}
{14,17,21,24,31,37}
{14,17,24,27,34,37}
{15,18,22,25,32,38}
{15,18,25,28,35,38}
{16,19,23,26,33,39}
{16,19,26,29,36,39}
{17,18,19,57,58,59}
{17,18,19,77,78,79}
{17,18,57,58,77,78}
{17,18,57,59,78,79}
{17,19,57,59,77,79}
{17,19,58,59,77,78}
{18,19,57,58,77,79}
{18,19,58,59,78,79}
{21,23,41,43,81,83}
{21,24,27,31,34,37}
{22,25,28,32,35,38}
{23,26,29,33,36,39}
{24,25,26,44,45,46}
{24,25,26,84,85,86}
{24,25,44,45,84,85}
{24,25,44,46,85,86}
{24,26,44,46,84,86}
{24,26,45,46,84,85}
{25,26,44,45,84,86}
{25,26,45,46,85,86}
{27,29,47,49,87,89}
{31,32,33,61,62,63}
{31,32,33,91,92,93}
{31,32,61,62,91,92}
{31,32,61,63,92,93}
{31,33,61,63,91,93}
{31,33,62,63,91,92}
{32,33,61,62,91,93}
{32,33,62,63,92,93}
{34,36,64,66,94,96}
{37,39,67,69,97,99}
{41,44,47,51,54,57}
{41,44,47,61,64,67}
{41,44,51,54,61,64}
{41,44,54,57,61,67}
{41,47,51,54,64,67}
{41,47,51,57,61,67}
{42,45,48,52,55,58}
{42,45,48,62,65,68}
{42,45,52,55,62,65}
{42,45,55,58,62,68}
{42,48,52,55,65,68}
{42,48,52,58,62,68}
{43,46,49,53,56,59}
{43,46,49,63,66,69}
{43,46,53,56,63,66}
{43,46,56,59,63,69}
{43,49,53,56,66,69}
{43,49,53,59,63,69}
{44,45,46,84,85,86}
{44,47,51,57,61,64}
{44,47,54,57,64,67}
{45,48,52,58,62,65}
{45,48,55,58,65,68}
{46,49,53,59,63,66}
{46,49,56,59,66,69}
{51,54,57,61,64,67}
{52,55,58,62,65,68}
{53,56,59,63,66,69}
{57,58,59,77,78,79}
{61,62,63,91,92,93}
{71,74,77,81,84,87}
{71,74,77,91,94,97}
{71,74,81,84,91,94}
{71,74,81,87,94,97}
{71,77,81,87,91,97}
{71,77,84,87,91,94}
{72,75,78,82,85,88}
{72,75,78,92,95,98}
{72,75,82,85,92,95}
{72,75,82,88,95,98}
{72,78,82,88,92,98}
{72,78,85,88,92,95}
{73,76,79,83,86,89}
{73,76,79,93,96,99}
{73,76,83,86,93,96}
{73,76,83,89,96,99}
{73,79,83,89,93,99}
{73,79,86,89,93,96}
{74,77,81,84,91,97}
{74,77,84,87,94,97}
{75,78,82,85,92,98}
{75,78,85,88,95,98}
{76,79,83,86,93,99}
{76,79,86,89,96,99}
{81,84,87,91,94,97}
{82,85,88,92,95,98}
{83,86,89,93,96,99}
{11,16,19,23,26,27,73,74,77,81,84,89}
{11,16,19,33,34,39,73,74,77,91,96,97}
{11,16,23,26,33,34,73,74,81,84,91,96}
{11,16,23,27,34,39,73,74,81,89,96,97}
{11,18,25,27,32,34,42,44,55,57,61,68}
{11,19,23,27,33,39,73,77,81,89,91,97}
{11,19,26,27,33,34,73,77,84,89,91,96}
{13,14,17,21,24,29,43,46,47,51,56,59}
{13,14,17,31,36,37,51,56,59,63,64,69}
{13,14,21,24,31,36,43,46,51,56,63,64}
{13,14,21,29,36,37,43,47,51,56,64,69}
{13,17,21,29,31,37,43,47,51,59,63,69}
{13,17,24,29,31,36,46,47,51,59,63,64}
{13,18,25,29,32,36,75,79,82,86,93,98}
{14,17,24,29,36,37,46,47,56,59,64,69}
{15,17,22,24,31,38,71,78,85,87,92,94}
{15,19,22,26,33,38,45,49,53,58,62,66}
{16,19,26,27,34,39,74,77,84,89,96,97}
{21,24,29,31,36,37,43,46,47,63,64,69}
{23,26,27,33,34,39,81,84,89,91,96,97}
{41,44,49,53,54,57,71,76,79,83,86,87}
{41,44,49,61,66,67,83,86,87,93,94,99}
{41,44,53,54,61,66,71,76,83,86,93,94}
{41,49,53,54,66,67,71,76,83,87,94,99}
{41,49,53,57,61,67,71,79,83,87,93,99}
{42,46,55,59,63,68,73,78,85,89,92,96}
{44,49,53,57,61,66,71,79,86,87,93,94}
{44,49,54,57,66,67,76,79,86,87,94,99}
{45,47,51,58,62,64,75,77,82,84,91,98}
{53,54,57,61,66,67,71,76,79,93,94,99}

Now we understand a bit more why it didnt have a 20