Some Logical Processes in Monster PuzzlesEM Easter Monster and the SK LoopThe original SK loop from Easter Monster uses 16 base sets and 16 cover sets to make what is logically equivalent to a continuous nice loop. All 13 eliminations occur because all the cover sets are rank 0.
16 Sets = {1267R2 1267R8 1267C2 1267C8}
16 Links = {56n2 8n4 28n5 2n6 45n8 1b37 2b19 6b37 7b19}
13 Eliminations
FM Original Loop-1 and BB (bi-bi) SetsThe first and easiest to find FM loop uses 3 floors of digits 136. It is the simplest example of a BB pattern with the first bi- pair in r5c46 (strong sets) and the second -bi pair in r4c2 and r6c8 (weak sets). The weak sets are highlighted black because they are rank 0 and can thus cause eliminations. The weak links in bi-bi groups are almost always rank 0, which is often not true of other weak links in the pattern.
11 Sets = {1C258 3C258 6C258 5N46}
14 Links = {1r159 3r259 6r158 4n2 6n8 136b5}
2 Eliminates = r4c2<>24
FM Applying ttt's AUR SolutionThis is ttt's first solution AUR
here that uses digit levels 1 and 6 and an AUR in r35c46. that combines with the loops on levels 1 and 6 to force 3r5c46 true, which eliminates the 6 3s in sets 3r5 and 3b5. Note: that ttt's AUR requires a weak cell set in r1c5 (which is in his diagram!) The BB pattern is not functional because there is no loop or other logic in the level 3 floor thus BB sets r4c2 and r6c8 are not rank 0.
10 Sets = {16R3 1C258 6C258 5N46}
15 Links = {1r159 3r5 6r158 4n2 1n5 6n8 6b1 1b3 136b5}
1 AUR = (16)r35c46
6 Eliminates = r4c45<>3, r5c278<>3, r6c5<>3
FM Combining ttt's AUR and Original Loop 1If the AUR solution is combined with the original loop, several more eliminations appear including those in r4c2<>24 and 5 new digit 3 eliminations. The 5 new digit 3 eliminations occur because the AUR forces the entire level 3 floor to rank 0, as noted by the black cover-set highlights. There are now 13 eliminations.
13 Sets = {16R3 1C258 3C258 6C258 5N46}
17 Links = {1r159 3r259 6r158 4n2 1n5 6n8 6b1 1b3 136b5}
1 AUR = (16)r35c46
13 Eliminations = r2c14<>3, r4c2<>24, r4c45<>3, r5c278<>3, r6c5<>3, r9c467<>3
FM Loop 2, Additional ConstraintsThe AUR above uses 2 row base sets 16R3 as part of its strong inference. The original FM Loop-2 below uses 6 column bases sets and 6 row base sets that include 16R3 used by the AUR. This loop eliminates 24r4c2 and 6 additional candidates in levels 3 and 6. The extra eliminations occur because 4 extra cell link sets in r37c46 introduce new constraints that make levels 3 and 6 rank 0. These 4 cell sets are not used to cover the logic and are not considered cover sets.
14 Sets = {136R3 136R7 136C2 136C8 5N46}
21 Links = {136r5 136c4 136c6 4n2 37n4 37n6 6n8 1b37 3b19 6b19}
8 Eliminations = r1c3<>6, r2c1<>3, r4c2<>2, r4c2<>4, r5c3<>6, r5c7<>3, r8c9<>6, r9c7<>3
FM Full Loop SolutionWhat happens if the AUR is integrated with the FM Loop 2? The initial answer is nothing unless all of the Loop 1 base sets are added, which will cause 15 eliminations.
However, Loop 2 defines an inner rectangle whose corners are r37c46. This rectangle also encloses 4 possible AURs. When all 4 AUR are added to the Loop 2 logic the result now produces 32 eliminations all based on the original 14 base sets, and no further integration produces additional eliminations. This pretty much destroys FM .
14 Sets = {136R3 136R7 136C2 136C8 5N46}
27 Links = {1r5 3r456 6r456 136c4 136c6 4n2 6n8 1b37 3b14569 6b14569}
4 AURs = (13)r35c46 + (16)r35c46 + (13)r57c46 + (16)r57c46
32 Eliminations
Platinum BlondPlatinum Blond makes a good comparison because it is both different and difficult. It does not share the same symmetry as many other puzzles however, like Golden Nugget, it can be morphed to a more symmetric form. Platinum Blond is also short because it uses floors 678 making it very short in 3D images. Platinum Blond also has some surprises.
PB Initial LoopAn initial search on PB shows a loop similar to Golden Nugget with a pair of strong cell sets (BB pattern) in box 3. However, it has 3 weak cell sets that are all rank 0 and cause eliminations. This is perhaps a bi-tri. Once major difference is the pattern uses a digit 4 base set. This loop makes 6 eliminations.
12 Sets = {679R3 679R4 4679R7 12N7}
15 Links = {6c157 7c167 9c257 47n8 7n9 679b3}
6 Eliminations = r4c8<>234, r7c8<>231
PB Initial Loop MorphPlatinum Blond is morphed to a form similar to FM and 4 AURs are added to the same locations that were used in FM.
PB Full Solution, 36 EliminationsA search discovers a solution of the same type as the FM full loop solution above, which uses all the same logical methods. This results in 36 eliminations.
18 Sets = {679R2 679R5 4679R8 679C3 679C7 46N5}
29 Links = {7r7 4c4 6c158 7c459 9c2458 28n4 8n6 4b8 6b357 7b14589 9b15689}
4 AURs = (67)r46c35 + (69)r46c35 + (67)r46c57 + (69)r46c57
36 Eliminations =
r2c4<>2, r2c4<>3, r2c4<>4, r2c4<>6, r2c5<>7, r2c5<>9, r3c1<>7, r3c5<>7,
r4c1<>7, r4c2<>9, r4c5<>6, r4c8<>9, r4c9<>7, r5c4<>7, r5c4<>9, r5c6<>7,
r5c6<>9, r6c1<>7, r6c2<>9, r6c4<>7, r6c4<>9, r6c5<>6, r6c8<>9, r6c9<>7,
r7c4<>7, r7c9<>7, r8c4<>2, r8c4<>3, r8c4<>6, r8c5<>7, r8c5<>9, r8c6<>1, r8c6<>6, r9c4<>9, r9c5<>9,r9c8<>9
PB What is This? 17 More EliminationsHowever, the Platinum Blond story is not finished, as perhaps that of other monsters. The following 17 eliminations are from completely different logic, which is also not based on floors. It has some resemblance to and SK loop because it is rank 0 with equal numbers of base and cover sets however, it is geometrically different from an SK loop and cannot be morphed into one.
It is made of from hidden and naked AALS and one additional set.
1) A hidden row AALS (4 rows with digits in 6 columns) in row 1.
2) An AALS in row 3. (digits 123467)
3) An AALS in row 7 (digits 123678)
4) An AALS in row 9 (digits 12389)
5) A box set 8B9.
16 Sets = {1235R1 3N3467 7N3467 9N467 8B9}
16 Links = {467r3 678r7 89r9 1c67 2c47 3c34 1n19}
17 ELiminations = r1c9<>4, r2c4<>23, r3c15<>7, r3c2<>4, r3c58<>6,
r4c3<>3, r6c7<>12, r7c9<>7, r8c4<>23, r8c6<>1, r9c58<>9