I Don't Think this Loop is Loopy at all!

Here's where you can play Parity Loop by Lithium-Ion: app.crackingthecryptic.com/0t...
Normal sudoku rules apply. All even digits in the grid except the even digits in the center box form a single, closed loop that moves orthogonally through the centers of cells, and does not branch, overlap, or touch itself (not even diagonally). A digit in a circle indicates exactly how many circles contain that digit. Digits in a cage sum to the value in the top left corner. Digits in a thermometer must increase from the bulb end.
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Hi! We're Simon Anthony and Mark Goodliffe, two of the UK's most enthusiastic puzzle solvers. We have both represented the UK at the World Sudoku Championships and the World Puzzle Championships. We're also "cryptic crossword" aficionados. Mark is the twelve-time winner of The Times crossword championship and twice Senior World Champion in Sudoku, and Simon is the former record holder for most consecutive correct solutions to The Listener crossword. We hope we can help your puzzle solving while also introducing you to some of the world's best puzzles.
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▶ Contents ◀
0:00 Theme Music & News around the channel
1:35 Rules of today’s puzzle
2:41 Start Of Solve - Let's Get Cracking!

[@Lithium-Ion-]

Hello guys, Lithium-Ion here. Many thanks for the feature. Mark followed the intended solve path from the beginning to the end. It was amazing to see how fast he looked at the right places at the right time. In particular, directly going for the parity loop instead of counting circles in the first place was the right intuition 🙂 I am very sorry that some people misunderstood "except the even digits in the center box" and thought it was "except the even digits in the center of each box". Nevertheless, I hope you enjoyed the puzzle.

[@daanperelachaise]

Great setting! I really enjoy 'lean' puzzles that have short rulesets and few givens, in which every clue serves a concise purpose. It flowed very nicely, Very elegant!

[@mbakenemdusink9757]

Wow, what a beautiful debut here! Thanks, loved it very much, hope to see more of your puzzles

[@dinane]

I actually counted circles first and discovered that it was not possible to have 4 odd circles due to the layout of box 7. Did all of that before color coding. But that led to the same path after that point. It was a lovely puzzle. Thank you.

[@jeffmoore688]

I also started with circles and deduced that they must contain 9, 6, 4, 2, and 1 due to the geometry of the puzzle. This forced the loop in box 7 and I continued from there.

[@jeffmoore688]

I'm also wondering if I can extend the method I used for determining the set of circled digits to other counting circle puzzles. Not sure what conditions are necessary

[@gordonglenn2089]

"You just put '[X] in a circle!!" I shouted. And it took a full 2 minutes before he spotted it!

[@57thorns]

Yes, that was rather painful. But there is so much information overload having to explain and show every step, and double checking that it is logical, it is easy to forget the premises occationally.

[@vulkandrache1928]

@@57thorns I would like to disagree with this. After the loop has been painted the only thing that matters is wether 9 is in a circle or not.

[@emilywilliams3237]

I especially like that the puzzle does not simply become two puzzles, even and odd, but other considerations stay in play until the very end. I loved the video - thanks, Mark.

[@FranklinCollins-re1he]

Took me 45 minutes! The speed at which you created the loop was incredible! It took me about half my solving time just doing that; I had none of your intuitions and it took me ages to get my head around how the loop would have to work. Once I did it was fairly simple to fill in the digits.

[@Timlagor]

I thought i was slowed by thinking the center of all the squares had to be even but maybe it actually helped me see how things worked

[@bertbergers9171]

Jikes, i messed up made an assumption in box 5, messing up my coloursceme, then getting nowhere from 50-75 minutes into the solve, and seeing a deadly pattern on the even loop, i started over. Watched Mark a bit to get the confidence that my start wasn't wrong at all, i discovered soon where i had to leave ambiguity instead of my earlier assumption. From there on out it was plain sailing in 22:02 for a total solve time of 97 minutes. I liked the idea very much puzzled a bit on the circles and how many evens there had to be in the circled numbers. That did help me a lot in starting the loop in box 7. Thank you Mark for the video and Lithium-Ion for a nice puzzle!

[@Xiuhtec]

Took me about 22 minutes just for the loop and 22 more to also solve the values, Mark doing both in 21 just astounds me but to be expected of such an expert. Beautiful puzzle!

[@Gonzalo_Garcia_]

15:24 for me. Very nice puzzle!

[@chocolateboy300]

I finished in 46 minutes. It took me some thought, but I realized that drawing the loop was very restricted on the edges. Using that, the loop practically drew itself. Then, I easily figured out the odd circles clue and completed it. Great Puzzle!

[@MattYDdraig]

34:29 I was a bit slow whittling down the circle possibilities, but the combo of those and the loop eventually clicked for me and this had some really clever trickery to it.

[@markp7262]

14:50 finish. A very fun little puzzle, excellent!

[@jdyerjdyer]

Easy and nice logical flow. Surprised to see that it had different paths for the solution. The logical path felt forced to me, but I missed a few logical deductions that might have made the solve easier, yet again. Also, there was a place or two where I think I used an easier logical argument than Mark. Either way, fun. :)

[@gatlygat]

I thought i absolutely blagged this, but on reflection i think it was intuition being half a step ahead of reasoning. Pretty wild ride, expecting the logic cops to catch me and book me, but instead making it all the way to the border and freedom. 22 minutes. Loved it!!

[@roccov3614]

That was immensely enjoyable. Every trick realization was a joy. Well done.

[@anaayoung9142]

Amazing solve Mark!! 🎉

[@danitajaye7218]

Very Well Done!!!! Fun video, excellent solve!

[@piarittersporn]

Very beautiful and enjoyable puzzle.

[@grahamania]

00:20:15 for me. Great puzzle. Not hard, but very fun! Kind comment.

[@benjaminrealy5661]

16:49. First quickly determined the loop had to include the outer edge of the non corner boxes and used the cages to fulfill the loop. So 3 even. Slow down point was not realizing 8 couldn't be a circle but once I did I knew the two odd circles must sum to 10 so knew where the 5 went in box 7. And smooth sailing from there.

[@osgrim7078]

Lovely puzzle! Fun to do first (30mins) and then nice watching the solve - as I suspected there were a few ways to break in and make progress through the puzzle (some puzzles follow a pretty strict path!).

[@subinmdr]

Took me 65 minutes. Spent 40 minutes on figuring out the loop (partly because I misread the rules as "the loop must pass through the center of each *box*" 🤦‍♂️). The best part was how the 5 in box 7 led me to working out the digits in the odd circles. Great puzzle!

[@NevelWong]

Same. I honestly still don't get it.What center of what box? I just ignored that rule after a while.

[@soulgorden]

A solid,clever solve

[@TurquoizeGoldscraper]

31:18 for me - got stuck figuring out where the 3 goes in box 2.

[@srwapo]

26:57, felt like I was stuck figuring out how to solve the odd digits for most of that time, thought I'd have to look at the video, then just figured out what to do and finished the puzzle in like 5 minutes.

[@roccov3614]

My first realization, Mark didn't mention. The thing I noticed is, of the 3 possible paths that the loop could take between squares, it could never pass through the line closest to the center square.

[@janeflett4971]

Cute puzzle

[@stevieinselby]

"And that's because ... I just feel it can't" ... good unimpeachable logic there 😂

[@Ray-iu7hg]

20:07 - took a while to figure out the break in.

[@victorfinberg8595]

45 min for me, and glad of it

[@butterfly17095]

I'm still relatively new to variant sudoku and have been struggling with this one. Around 16:30, Mark talks about how 1 and 3 have to be in the rightmost column in box 3 but I don't understand the logic of why r3c9 can't be 9. I know it doesn't work because I tried it and it broke the puzzle lol, but I don't know why. Can anyone explain that for me?

[@crisx0573]

was because the thermo rule, had to be incrementing.

[@butterfly17095]

​@@crisx0573but the cell I mentioned isn't part of the thermo? I was thinking the thermo could potentially be 1 3 5 7, then 9 outside of the thermo

[@robertcousins2274]

21:56 for me

[@MatthewMorris6148]

7:15:59 In my defense, I fell asleep.

[@AndreAy1975]

Solved it with a few hints from the video.

[@mudscuffer]

41:16 for me, as I made a mistake on my first loop and ended up clearing the grid.

[@LednacekZ]

22:52 today. horrible solve. spent at least 10 minutes trying to figure out if it is 19 or 37 in the odd circles. In the it was so obvious, but took me long to get it.

[@srwapo]

Same!

[@Kirbyfan87827]

Finished in 15:58 with much help from the video - there was no way I'd be able to figure out both the loop and counting the circles on my own.

[@tonybrooms]

00:45:47

[@alienrenders]

19:54 for me. A little too simple for my taste. But interesting idea nonetheless.

[@pattoner8398]

I think I cheated by making a huge assumption that worked: if each box had a different number of circles, they'd obviously be the numbers 1-9 and one box would have exactly one circle and it would be a 9. The assumption I made started there but there are three boxes with one circle so they're all 9. If all the numbers were represented there would be a box with two circles and that would have 8 and 9. For there to be an 8 in a circle there would have to be only one box with one circle and the rest would have minimum two circles. Since that's not the case 8 is out. Same for 7. Next we have two boxes with two circles, so those would be 9 and 6, then we can't do 5 because we had two boxes with 6, so the boxes with three circles contain 9 6 and 4. Again there are two of those boxes so 3 is out leaving us with 9642 in the four circle box and 96421 in the 5 circle box. Can someone prove if this should always work for this type of puzzle?

[@karlrado]

That’s exactly how I did it. I didn’t determine the loop early on, but the loop constraint certainly helped place the even digits after narrowing down the candidates for the circles . I can’t prove if my solve that way was valid because I may have gotten lucky. But it did “feel” like a correct solve.

[@RichSmith77]

You weren't forced to have 9 appearing in circles just by the disposition of the circles. You could have had 23467 arranged in the circles too, if there were no cages or thermos in the puzzle. Just because there's a circle in every row, column and box, doesn't necessarily mean there has to be a 9 in circles. For example, take Mark's solution, swap 9s for 7s and 1s for 3s. Then swap two of the 7s for two 3s. (Delete everything that's not in a circle).

[@pattoner8398]

​@@RichSmith77 I know that's why I feel like I cheated but I've seen puzzles with that constant before and used that logic (they were more constricted on what digits they could be) I just thought this was another interesting way to look at it

[@LukeSumIpsePatremTe]

20:45

[@theredstoneengineer6934]

14:07 for me

[@michaellautermilch9185]

21:14 - that's how you do the 3 in the corner song, like a dying broken record. How fitting for someone who wants to mock God who gives life.

[@mussoletart8485]

31 minutes, really proud of myself.

[@MrRockapoodle72]

I dont understand the doesnt touch itself diagonally rule. You literally cant make a closed loop without it touching diagonally

[@steve470]

The intended meaning (for this and any puzzle that involves a loop/snake not touching itself diagonally) is that cells on the loop/snake that aren't orthogonally adjacent to another loop/snake cell can't touch diagonally. In other words, the loop/snake is allowed to turn (i.e., you can go from r4c3 to r3c3 to r3c4, despite the fact that r4c3 and r3c4 are diagonally adjacent), but one part of the loop/snake can't touch a different part diagonally (i.e., you can't go from r4c2 to r4c3to r5c3 and then later on go from r3c5 to r3c4 to r2c4, because different parts of the loop/snake are now touching diagonally at r4c3 and r3c4).

[@RichSmith77]

I find it helpful to think of the inside of the corner, where the two cells would touch diagonally where the snake turns, as being a single point of the snake. The rule isn't that cells on the snake cannot touch (not even diagonally). It's the entity that is the entire snake that may not touch itself, not even diagonally. After all, if it was a rule about individual cells not touching, not even diagonally, then the rule would be broken by every pair of adjacent cells along a straight section of snake too. They're also cells in the snake, and they're touching orthogonally, not even diagonally. For something to be said to be touching itself, two *distinct* points on its surface need to come into contact. (If you bend your elbow at 90⁰, does that mean you're now touching yourself? How about if you bring your finger into contact with your nose?)

[@Timlagor]

I'd have started much better if I hadn't interpreted the rules as saying the center of each box was even

[@RoderickEtheria]

I just stumbled across the answer by drawing a possible loop.

[@clovr3430]

Found the loop quite hard to get correct. I ended up with your loop but reversed and ended up here

[@esti369]

Me: “He’s put a [X] in a circle. He’s going to need to see that to get much further.” Mark (2 seconds later): “Oh I’ve put a [X] jn a circle!” Me: “There we go!” Lovely puzzle. Good solve! 👍🏻

[@angec9908]

First like!😊

[@Paolo_De_Leva]

_"not even diagonally"_ That's meaningless and useless if you refer to a loop formed by a *line* (as opposed to a one-cell wide loop). A _"loop that moves orthogonally through the centers of cells"_ is a *line* composed of horizontal or vertical segments. There's nothing diagonal in it. It can touch or intersect itself *at a point* (the center of a cell) or along a non-diagonal segment. It can never touch itself _"diagonally❗"_ On the contrary, a one cell wide loop, formed by _"orthogonally connected cells",_ may be said to touch itself _"diagonally"_ when two of its cells are connected by one of their corners (referring to the diagonal line from one cell center to the other). This is horrible terminology, but at least it is widely used and it is not totally nonsensical. In short, my point is that the rule is described ambiguously. At least one part of it is measliding. Which one? It takes a while to understand that it refers to a *one-cell wide* loop, and I believe you need to start solving before understanding.

[@Paolo_De_Leva]

By the way, this was a beautiful sove by Mark 👏👏👏👏👏👏

[@bruceh8043]

27:02 for me and solver #3475.