Overview ======== - which type is which - the zippers - the arrows - the geometry trick (spoiler - phistomefel) - the whisper trick (rows 2 and 8 cannot have too much on common) - leverage from columns 2 and 8 - whisper redux - pivot from column 8 to rows 2 and 8 and the corners - speedrun Which type is which ------------------- Zipper being entropy sounds hard, as the middle is high, and that's mid-arrow. Those arrows can't have a 0mod3 on its shaft, so would all be 9={7,8}+{2,1} in alternating order. You now can't place an 8 in the 9=7+2 corners. => Zippers are modular and arrows are entropic The zippers ----------- Being modular, the zippers have [36] in their middles. The ones 3s are flanked by [12] pairs, the ones with 6s are flanked wither with [15] or [24] pairs. [1245] are useful pencil-marks The arrows ---------- The two arrows with a 6 (on a 1-6-5 or 2-6-4 zipper) on them, wherever they are, can't be {7,8}=6+{1,2}, because [12]s get reflected back from teh zippers opposite, and so neither end of our zipper can be either of 1 or 2. => Two of teh arrows are 9=6+3 The two arrows with a 3 (on a 1-3-2 zipper) must house the 6 in their box at their tip, so are 9=3+6 The geometry trick ------------------ Thie corner boxes corners, and the regions between the kropke dots are an exploded Phistomefel ring. So it 9999, 33, 66, 12 twice, and two of either 15 or 24 are in boxes 1,3,7,9, then exacty the same is in rows/columns 2 and 8. The 33 and 66 are already in both sets, so the regions in question are only 12 cells each. Note - there four 9s, so there's one on each run of three, and it can't be on the black kropke. r[28]c[456] and r[456]c[28] can all be pencilled [12459], as we'll immediately be able to start cleaning that up. For example, you can't put an 8 in r8c3, as 8x4 forces the white kropke to be 1+2, and thus puts the 9 in r2c9 and an 8 in r2c3. This makes the black kropke 1x2 or 2x4, and thus removes 2 from the rest of row 8. This makes the white kropke not 1+2, removing 1 from itself The same does not apply to column 8, as there's no restriction on the position of the 9 in column 8. However, it does apply to row 2. The whisper trick ----------------- Rows 2 and 8 now have enough restriction to decide the [36] issue: If the 3s were both next to black kropkes, the 3-4-2-[15]-9-8 pattern would be forced in both rows. However, the [29] xwing (in fact any [{123}{789}] xwing) in columns 4 and 6 breaks the GW/X pattern in box 5. => 3s in r2c2 and r8c8, 6s in r2c8 and r8c2 => [12] pairs sandwich the 3s on zippers, and [12] comes out of other cells in their box. => column 2's black kropke has the 4 on it, so it's not elsewhere in the column => the white kropke can't have a 5 on it either leverage from columns 2 and 8 ------------------------------ r7c2 can't be 1, as it forces [24] onto three other cells in teh box There's now a [248] tiple, so r5c2 can't be 2. r3c8 likewise. We can do more there - r3c7 can't be 2, as the [12] in r7c7 makes both r2c7 and r1c9 4. Even more - r7c2 can't be 2, as that forces the zipper to be 1-6-5, but that 2 bounces off the [12] in r7c7 to make r2c7 [24], and thus box 3's zipper 1-6-5. That means there are no 4s in the phistomefel, which breaks r4c2. => the 8 in r7c2 places teh +9 and 4x2 above it => the 4 in r3c2 forces the zipper in box 3 to be a 1-6-5 => r3c7 is restricted to [28], and r4c8 becomes [19] Whisper redux ------------- in row 4, the X can't be [19] (breaks r4c8), [28] (breaks r4c2), or [46] (breaks GW), so is [37] r5c[46] is either a [12] pair or an [89] pair. However, boxes 1 and 7 provide the 1s for columns 1 and 3, so r5c2 is 1 => r5c[46] is an [89] pair and r4c[45] is 7x3 pivot from column 8 to rows 2 and 8 ----------------------------------- Box 6/column 8's 9 is forced into r4c8 => r3c8 is 8 Likewise, the 1 can only be on r6c8, making r6c5 a 2, and r4c6 a 1 => box 2/row 2's 1 must be in r2c5 => the [24] pair in r2c[67] makes the white kropke 8o9 => r5c4 is 8 and r5c6 is 9 => box 8/row 8's 9 must be in r8c5, and that's the 9s completed The [45] pair created in r8c makes the black kropke a [12] pair. You're basically in speedrun mode now: r8c1 is a naked 7, which windmills to boxes 1 and 3, and forces 8 into r8c9 r9c2 is a naked 5 => the box 7 zipper is a 2-6-4 => all corner cells resolve => the black kropke on row 2 becomes 2x4 => the white kropke on row 8 becomes 4o5 => box 9's 7 and 4 slot immediately in => column 8's 2 and 5 follow => column 9 resolves 4/2/3 => column 1 resolves 8/3/4 => column 7 resolves 6/7/8 => column 3 resolves 5/6/7 Row 7 resolves 1-5-3 => row 3 resolves 3-7-5 => row 6 resolves 5---6 => row 1 resolves 4-6-8 => row 9 resolves 6-8-7 proof of uniqueness: https://sigh.github.io/Interactive-Sudoku-Solver/?q=.Zipper%7ER1C1%7ER2C2%7ER3C3.Zipper%7ER1C9%7ER2C8%7ER3C7.Zipper%7ER7C3%7ER8C2%7ER9C1.Zipper%7ER7C7%7ER8C8%7ER9C9.Arrow%7ER1C3%7ER2C2%7ER3C1.Arrow%7ER3C9%7ER2C8%7ER1C7.Arrow%7ER9C7%7ER8C8%7ER7C9.Arrow%7ER7C1%7ER8C2%7ER9C3.Or.And.Modular%7E3%7ER7C3%7ER8C2%7ER9C1.Modular%7E3%7ER7C7%7ER8C8%7ER9C9.Modular%7E3%7ER1C9%7ER2C8%7ER3C7.Modular%7E3%7ER1C1%7ER2C2%7ER3C3.Entropic%7ER3C9%7ER2C8%7ER1C7.Entropic%7ER1C3%7ER2C2%7ER3C1.Entropic%7ER7C1%7ER8C2%7ER9C3.Entropic%7ER9C7%7ER8C8%7ER7C9.End.And.Entropic%7ER9C1%7ER8C2%7ER7C3.Entropic%7ER9C9%7ER8C8%7ER7C7.Entropic%7ER1C1%7ER2C2%7ER3C3.Entropic%7ER3C7%7ER2C8%7ER1C9.Modular%7E3%7ER9C7%7ER8C8%7ER7C9.Modular%7E3%7ER3C9%7ER2C8%7ER1C7.Modular%7E3%7ER1C3%7ER2C2%7ER3C1.Modular%7E3%7ER7C1%7ER8C2%7ER9C3.End.End.WhiteDot%7ER6C2%7ER7C2.WhiteDot%7ER2C3%7ER2C4.WhiteDot%7ER3C8%7ER4C8.WhiteDot%7ER8C7%7ER8C6.BlackDot%7ER8C3%7ER8C4.BlackDot%7ER3C2%7ER4C2.BlackDot%7ER2C6%7ER2C7.Whisper%7E5%7ER5C4%7ER6C5%7ER5C6%7ER4C5.Whisper%7E5%7ER4C5%7ER5C4.X%7ER4C4%7ER4C5