Thermo miracle methods

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Firlz
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Thermo miracle methods

Post by Firlz » Fri 11 Dec, 2020 4:58 am

Spent literally the better part of nine hours solving https://cracking-the-cryptic.web.app/sudoku/bDQtpNhg4h

I got a solution that as far as I can tell follows all the rules correctly, but I lost something like SEVEN+ hours just on what was basically the second step.

Having only needed the better part of an hour to solve the other MIracle sudokus that were linked in those videos, it was very depressing for it to take so much time to do this one.

I was immediately able to see that the 3-long thermo bulb had to be a 1-5. I noticed that if it was 5 it had to be 5 - 7 - 9. I also noticed that there has to be a minimum increase of two between each digit on the thermo.

The only thing I could think at that point was to attempt to prove or disprove that it was possible for 5-7-9 to work.

Multiple copies of the puzzle to have "heat maps" I could use to derive things between each other using coloring I was able to deduce that if the bulb and next number were separated by exactly 2, then the number 1 less than the bulb number could not possibly be placed. This also showed me that if the bulb was 2 less than the next number exactly which squares the remaining 8 copies of the bulb number had to be in. That told me it had to be 1-4, and that if 2-4 the next number had to be 3 or more larger.

So just to see if anything else became apparent I plugged 1 and 3 in, and just went to see where the rest of the puzzle went. I got lucky that was the correct thermo combination and everything went fine from there.

But there HAS to be a better way to figure out what is going on with the thermos, the method I used made the second copy superfluous, which means there must be some trick relating the two together to speed up the process. But I just can't for the life of me figure out what the intended procedure was for this puzzle.

Is there some other trick with thermos I don't know and I am just too new to sudoku variants to notice and it was my lack of experience with other puzzles that made me have to go through such an arduous method?

Thanks

GarethMoore
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Re: Thermo miracle methods

Post by GarethMoore » Fri 11 Dec, 2020 9:35 am

If you want people to help you'll need to explain the rules, since this clearly isn't just a regular thermometer sudoku. :)

I tried googling your link but it didn't bring up the video with the rules, and 'Thermo miracle' brought up a different cracking the cryptic puzzle.

detuned
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Re: Thermo miracle methods

Post by detuned » Fri 11 Dec, 2020 10:27 am

The trick with “miracle” aka anti-knight combined with no touch constraint is that there is one solution, up to dihedral symmetries of the grid (reflections/rotations) and permutations of the symbols (eg swapping all the 1s with all the 3s). To be honest, despite the hype, I think that “miracle” puzzles aren’t really miraculous armed with that knowledge, and aren’t really that much fun once you know the trick.

The thermos in this grid are there to fix exactly one dihedral symmetry and one permutation of symbols that works.

People have proved this trick by brute force (a computer search), one of these days I’ll finish a more elegant proof. One observation to get started with is that there are many overlapping groups of 8 cells in the grid which all have to contain different numbers.

Firlz
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Re: Thermo miracle methods

Post by Firlz » Fri 11 Dec, 2020 6:12 pm

@detuned, how do the thermometers fix the symmetry here? I think that's the part I don't understand.

I was able to brute force my way into a solution (with a bit of luck as one of my early "let's assume this" points ended up being quite useful) but the upper right thermo became completely superfluous as a result.

Is there some relationship between the two thermos (or even within each thermo) that limits the possible numbers in each? That's what I think I wasn't seeing.

detuned
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Re: Thermo miracle methods

Post by detuned » Fri 11 Dec, 2020 10:50 pm

Every solution to the anti-knight no touch constraint looks something like this:
Screen Shot 2020-12-11 at 22.40.46.png
Screen Shot 2020-12-11 at 22.40.46.png (159.62 KiB) Viewed 1265 times
All you need to do is permute the numbers attached to each colour, and possibly rotate by 90 degrees, until you find something that works. To suggest there was some "intended procedure" for this puzzle is probably incorrect. Many sudoku setters might tweak with an idea until they find a unique solution and leave it there. If you are lucky, the setter might also have tried testing the puzzle in the knowledge of a unique solution to try and verify some kind of logic that is to their own satisfaction (I think this is more or less how Sam set the recent killer-little killer prize puzzle on cracking the cryptic for instance)

I suppose if you were to look for further interactions between the two thermos, one observation is that in one orientation, those 5 cells contain 5 different colours, and therefore 5 different digits - in the other (where you rotate by 90 degrees) there is exactly one digit repeated. Maybe that's helpful.

One further thought - I'm not sure if your miracle constraint also includes non-consecutive. That helps reduce the numbers of permutations to choose from as well.

Firlz
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Re: Thermo miracle methods

Post by Firlz » Sat 12 Dec, 2020 10:44 am

Yeah, there is a no consecutive rule, which was the only reason it could have a unique solution as far as I could tell.

If I understand your diagram, any color can be any number (the example is just one possible way of filling it out for when the top left box is 1-9 consecutive) right?

Also, that mapping is also retained using reflections in addition to rotations, right?

detuned
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Re: Thermo miracle methods

Post by detuned » Sat 12 Dec, 2020 3:00 pm

My bad, I’m being a bit lax with the dihedral symmetries. There are normally a total of 8 (4x multiples of 90 degree rotations, 4 more reflections in the | - / \ axes) but I was probably getting a bit excited given that a 180 rotation in that grid is actually the same thing as a digit permutation (18)(27)(36)(45) which cuts things down to 4 representative symmetries to check, if I remember my maths correctly

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