Overview: ========= 1000 feet 789 147 entropy colouring modularity numbering 456 Bring it home View from 1000 feet ------------------- There's exact rotational anti-symmetry - rotate the board and every bulb maps onto a tip, and every tip maps onto a bulb. Therefore - *if there's a unique solution* - every number is opposite its sum-to-10 partner, and there's a 5 in the centre of the grid. However, I didn't tell you you could assume that, so, if you like the cslightly more challenging mode, please feel free to do this without that knowledge. Personally, I think you can assume uniqueness, but I think CtC prefer not to, so that's the walk-though I'll provide. I chose to colour entropy and pencil [147], [258], [369] modularity. It's probably possible to colour modularity and pencil entropy. I'm colourblind and can't imagine colouring both, but some nutters might like to try. Whatever works for you is the right way. There must be 789 thermos ------------------------- The entropic line through all the bulbs must have a high entropy at at least two locations along its length, and 8 or 9 are impossible, so there must be at least two 789 thermos. The entropic bulbs are mostly [147] ----------------------------------- The middling bulbs cant push another 7,8, or 9 into the entropic thermo-tips, so must be 4s, starting 456 thermos. The low bulbs can't push another 4-9 into the entropic thermo tips, so must be 1s, starting 123 thermos. All of the length-3 thermos can be pencilled [147], [258], or [369] Entropy colouring ----------------- We don't know what order they're in, but the thermos on the diagonal must each be monochromatic in entropy. This even applies to the short one in the corner. I'll refer to the two colours that occur in the corners as corner colours, and the one that appeasrs in the very middle the middle colour - that's a geographical middle, I make no reference to its value (because we donnot assume uniqueness, see above). Six box-corners immediately reveal their corner colours, three of each, four of which can be numbered, being the third of that colour. Between those six corners, the rest of those two diagonals can be coloured, as it's the ones that abutt the middle colour can only be the remaining corner colour. That's 8 more cells, four of which can be numbered. Three of each of the two corner colours point at r3c7 and r7c3, so that's middle colour. That permits their diagonals to be coloured too, using the same logic as prior. Only four of them can be numbered. Starting with the bulb on the eext diagonal out, the same logic applies to that diagonal too. No trivial numering along there yet. Row 5 reveals its two missig cells' colours and numbers. Columns reveal their full colourings, which reveal boxes full colourings, so liberally, that you can't keep up with numbering as you go - just paint them, we'll do numbers later in a proper sweep. Modularity numbering -------------------- With a similar approach to colouring, we can address numbering. Start with the corner colours, they're just screaming to be filled in everywhere. Note the length-2 thermos in boxes 3 and 7 resolve, and thus tell us teh length-2 thermos in boxes 1 and 9. The middle colour requires only the barest level of insight - that r2c1=r2c4, and r9c8=r8c6 - in order to achieve enough numbering to progress to the next phase. However, you might as well use r6c5=r5c3 or r4c5=r5c7 to fill everything with pencil marks. Resulution of which colour is [456] ----------------------------------- You have the two diagonal length-2 thermos saying: (middle-369) < [corner 369] in box 7 (othercorner-147) < [middle 147] in box 3 Therefore teh middle colour is the middling entropy, top-left corner colour's lower, the bottom right corner colour's higher. No speedun here, just replace low-[147] with 1, low-[258] with 2, mid-[147] with 4, etc. You might say most of it is speed-run, it's just been a speedrun in abstract colours and modular sets rather than a sppedrun of identified values. You did leave low-[369] to the end, I hope? Proof of uniqueness: https://sigh.github.io/Interactive-Sudoku-Solver/?q=.Thermo%7ER2C1%7ER1C1.Thermo%7ER3C2%7ER2C2%7ER1C2.Thermo%7ER4C3%7ER3C3%7ER2C3.Thermo%7ER5C4%7ER4C4%7ER3C4.Thermo%7ER6C5%7ER5C5%7ER4C5.Thermo%7ER7C6%7ER6C6%7ER5C6.Thermo%7ER8C7%7ER7C7%7ER6C7.Thermo%7ER9C8%7ER8C8%7ER7C8.Thermo%7ER9C9%7ER8C9.Thermo%7ER8C1%7ER7C2.Thermo%7ER3C8%7ER2C9.Thermo%7ER1C7%7ER2C7.Thermo%7ER8C3%7ER9C3.Entropic%7ER2C1%7ER3C2%7ER4C3%7ER5C4%7ER6C5%7ER7C6%7ER8C7%7ER9C8.Entropic%7ER1C2%7ER2C3%7ER3C4%7ER4C5%7ER5C6%7ER6C7%7ER7C8%7ER8C9.Entropic%7ER1C1%7ER2C2%7ER3C3%7ER4C4%7ER5C5%7ER6C6%7ER7C7%7ER8C8%7ER9C9.Entropic%7ER1C3%7ER2C4%7ER3C5%7ER4C6%7ER5C7%7ER6C8%7ER7C9.Entropic%7ER3C1%7ER4C2%7ER5C3%7ER6C4%7ER7C5%7ER8C6%7ER9C7.Entropic%7ER1C4%7ER2C5%7ER3C6%7ER4C7%7ER5C8%7ER6C9.Entropic%7ER4C1%7ER5C2%7ER6C3%7ER7C4%7ER8C5%7ER9C6.Entropic%7ER1C5%7ER2C6%7ER3C7%7ER4C8%7ER5C9.Entropic%7ER5C1%7ER6C2%7ER7C3%7ER8C4%7ER9C5.Entropic%7ER1C6%7ER2C7%7ER3C8%7ER4C9.Entropic%7ER6C1%7ER7C2%7ER8C3%7ER9C4.Entropic%7ER1C7%7ER2C8%7ER3C9.Entropic%7ER7C1%7ER8C2%7ER9C3.Entropic%7ER1C8%7ER2C9.Entropic%7ER8C1%7ER9C2