"All Hail Hypnorenban II" Renbans, entropies, and German whispers can be solved only up to sum-to-10 parity, only the black kropkes can distinguish high from low, so it makes most sense to solve for A-I rather than 1-9 until parity is known. However, it also makes sense to colour for entropy with 3 colours. The 3 renbans on row 1 must be [ABC]/[DEF]/[GHI] in some order. If box 1's component was [DEF], its entire dogleg renan would be [ABC] or [GHI], which must also connect on row 2 to something that's in the other set, which is impossible. => box 1 starts [ABC], be that high or low. However, we don't know whether box 2's or box 3's component is [789]. Column 1's whole renbans must include the two of [ABC] that aren't in r1c1, and obviously can't be [ABC], so one of them must be [BCD] and r1c1 is A. The other one is [EFG] or [GHI], but must contain a G, leaving the dogleg renban fully in box 1 as having an [EF] or [HI] pair in column 1. No matter which of the two it is, the kropke dot on row two and adjacent renban removes too many higher letters, and forces the rightmost renban in row 2 to be [ABC]. Hypothesizing it's [HI] puts the G on the kropke, so G=3 paired with 6, and A=9. However, don't renumber the letters, this quickly leads to a dead end. The D in row 2 pushes [EF] into box 2 along the renban, making row 1 [ABC]/[GHI]/[DEF] A D is pushed into r4c2, and therefore the [BCD] reban in column 1 goes in box 3. This makes it hard to put an A in row 9. If r9c1 is B, the A=9 must go, with a C=7, in the dogleg renban in box 9, but the kropke prevents those. If r9c1 is C, the dogleg renban is [AB], and its C is pushed round the corner, so r1c9 must be [AB] - imposs. If r9c1 is D, one of the straight renbans must be ABC, and we're left with the same situation as box 1, the very thing we're trying to resolve - the dogleg's either EF or HI. However this time it's easy to reolve, as [EF] forces the column 9 kropke to join an [ABC] to a [GHI], and [HI] pushes a G onto column 9 where it clashes with the renban in rows 2-4. => it's not [HI] in the dogleg in column 1 Given that it's a [EF], it can't be completed with a G, which pairs with a D, and makes the other renbans unsatisfiable. => The box 1 dogleg is [EF] in column 1 and D in r2c2, pairing to [GH] in r2c3. This [GH] pushes another of [GHI] into box 2 along the renban, so row 1 resolves [ABC]/[DEF]/[GHI]. The other dogleg renban in box 1 must be [GHI], but the I is in box 1. Column 1's other 2 renbans must now be [BCD] above [GHI] (The entropy line in column 3 must coexists with the [BCD] renban, so must contain A) The entropy line in column 2 must contain an A, B, or C, which must be external to box 4. Row 9 and column 9 are very similar in composition, and should be considered together If either starts with a G or H, then the renbans must be [ABC] and [DEF], leaving [GI] or [HI] in box 9 If they start with an I, they could be [ABC] with [DEF] leaving [GH], [ABC] with [FGH] leaving [DE], or [CDE] with [FGH] leaving [AB]. However, if one of them leaves [AB], or [DE], the other must too, and the dogleg renban in box 9 contains a duplicate. Therefore at least one starts with a G or H, and the other is a different member of [GHI], both have [ABC] and [DEF] as their renbans, and [GHI] is in the box 9 dogleg renban. Row 9 has the [ABC] before the [DEF] in order to mate at the kropke with [GHI] Column 9 already has an [EF] in r2c9, so the order's defined. A black kropke of [DEF]o[ABC] can be DoB (4o2), FoC (6o3), or FoB (4o8) A black kropke of [DEF]o[GHI] can be FoH (4o8), DoG (6o3), or DoH (4o8) These force an E into r9c6, and push I into column 9 of box 9 and out of r1c9. Box 5, being 3 renbans, must be [ABC]/[DEF]/[GHI], but row 3 rejects [GHI] and [DEF], and column 6 rejects [DEF], so can be fully coloured. This places D in r4c1, F in r4c9, E in r2c9, D in r3c9, F in r2c1, and E in r3c1 The ABC triple in box 5 and D in r4c1 prevent the renban in column 7 from being [ABC] or [BCD], but it could be C-E-D. However, hypothecating a C into r3c7 forces F into r9c7. The dogleg renban partly in column 7 needs to contain only values from [GHI], but there's no room in box 9 for those. => r3c7 is not C, it's F => r9c7 is D, and r9c5 is F The renban from r7c3 starting cannot continue [DEF], or two [ABC]s, so must be an F contining with two [GH]s. => r8c2 is E and r8c3 is D => the row 8 renban continues with a C. => r7c1 is I => r9c9 is I The dogleg renban in box 9 can only be [CDE], placing the D in r7c6 => the straight renban in column 7 cannot have an E, and is F with [GH] => r1c6 is F and r1c7 is I r8c8 is a naked F, and its renban may not continue with a G, so is [DE] in known order. This resolves the dogleg renban beside it into E above C. => r6c4 is F, r5c4 is E, and r5c5 is D, resolving the [DE] in row 1. => r5c2 is F => r4c3 is E, r5c3 is [GHI], and r6c3 is A => r7c9 is A, r8c7 is B, r2c7 is A, r7c2 is B, r9c3 is C, r9c2 is A, r9c4 is B, r4c4 is A, r1c2 is C, and r1c3 is B Sudoku places I above A in column 8 box 6, and the modular line demanding a B pushes resolution of Bs and Cs into boxes 3, 2, and 5. Sudoku also places A then I in row 8 box 8. That pushes Is into r6c5, r2c4, r5c3, and r3c2 Sudoku also places A in r3c6, C in r3c5, and either G or H in r3c4. However, that's a modular line, and only H goes. This resolves [GH]s in boxes 2 and 1, most importantly, making the box 1 kropke a [DH] pair, which can only be [48], so we can turn A->1, etc. The black kropke in box 9 resolves boxes 7, 9, and 3's [78] pairs, and across the box 1/4 renban reolves boxes 4, 5, and 6 78 pairs too. The black kropke in box 6 resolves boxes 6 and 4's [23] pairs. Proof of uniqueness: https://sigh.github.io/Interactive-Sudoku-Solver/?q=.Renban%7ER1C1%7ER1C2%7ER1C3.Renban%7ER5C9%7ER6C9%7ER7C9.Renban%7ER7C1%7ER8C1%7ER9C1.Renban%7ER2C6%7ER2C7%7ER2C8.Renban%7ER8C4%7ER8C3%7ER8C2.Renban%7ER3C7%7ER4C7%7ER5C7.Renban%7ER4C4%7ER4C5%7ER4C6.Renban%7ER1C7%7ER1C8%7ER1C9.Renban%7ER2C2%7ER2C1%7ER3C1.Renban%7ER9C7%7ER9C6%7ER9C5.Renban%7ER6C8%7ER7C8%7ER8C8.Renban%7ER4C2%7ER3C2%7ER3C3.Renban%7ER7C5%7ER7C4%7ER7C3.Renban%7ER6C4%7ER5C4%7ER5C5.Renban%7ER1C4%7ER1C5%7ER1C6.Renban%7ER2C9%7ER3C9%7ER4C9.Renban%7ER8C9%7ER9C9%7ER9C8.Renban%7ER9C4%7ER9C3%7ER9C2.Renban%7ER6C1%7ER5C1%7ER4C1.Renban%7ER2C3%7ER2C4%7ER2C5.Renban%7ER6C7%7ER7C7%7ER7C6.Renban%7ER5C6%7ER6C6%7ER6C5.Entropic%7ER5C2%7ER6C2%7ER7C2.Entropic%7ER4C3%7ER5C3%7ER6C3.Modular%7E3%7ER3C4%7ER3C5%7ER3C6.Modular%7E3%7ER3C8%7ER4C8%7ER5C8.Modular%7E3%7ER8C7%7ER8C6%7ER8C5.BlackDot%7ER2C2%7ER2C3.BlackDot%7ER9C7%7ER9C8.BlackDot%7ER5C9%7ER4C9.WhiteDot%7ER9C6%7ER9C7.WhiteDot%7ER5C4%7ER6C4