The Miracle Sudoku: Rubik's Cube

[@r.b.4009]

So I am watching this on the TV while my wife gets in and out of sleeping. After 25min she says "what he still didn't find a number?" And my response was "well, I didn't even understand the rules yet!" 🤣

[@hypotheticalhippo2327]

Whenever Simon takes on a Sudoku that includes a disjoint set rule, we need Mark in the background reading a book or something. Then whenever Simon gets stuck, just have him shout "Disjoint!"

[@krm1t]

Is that a suggestion for a live stream monetization? I.e. Simon live streaming a solve and then viewers can donate something like 2.44 euro (exactly) to have a small video clip pop up of Mark raising his head from the book and shout Disjoint.

[@bombmk2590]

I was screaming at the screen every time he put pencil marks in without checking them against the disjoint rule. :)

[@TheGeronimo2]

Simon randomly clicking the square that he should look at and asking where should he look now, then proceeding to somewhere else is what gets me every time :)

[@katiekawaii]

Haha every time!

[@neilkightley3451]

I wonder if colouring each disjoint box a different colour would help Simon spot them?... I doubt it! 🤣

[@maht0x]

we used to say "check the square under the mouse" but Simon seems to have stopped saying that, even though it's still true

[@ghandigoots]

Simon you are definitely not an inept human being! You might struggle with disjoint sets, but you are one of my favorite people and I bet 500,000+ people would agree with me

[@Anne_Mahoney]

+1

[@fredgoodyer4907]

It was genuinely such a sad moment, I had to pause the video to gather myself 😅

[@crashtextdummie]

I'm not usually one of the fabled shouters, but I did actually find myself shouting "DISJOINT! DISJOINT!" at the screen through large portions of the video this time around.

[@JohnDBlue]

I didn't shout, but there was a fair bit of frustration every time he looked at the 2 pencil mark in row 9 column 5 😅

[@mvanvoorden]

I was finding myself a bit frustrated as there was one deduction to be made that would've saved so much time: - The numbers on row 1 index all the 1s in their respective rows - The numbers on row 2 index all the 2s etc etc This way one could simply ask for each cell on row 1 where can the 1s go on row X and repeat for each row.

[@ghandigoots]

Disjoint sets really are his kryptonite. When he clicked on the 3 pencil mark in box r8c6 and asked where to look next, I shouted just a bit myself. He definitely understood the XYZ thing way better than me though and certainly IS NOT an inept human being! He's a wonderful human being. I'd hate to hear him shouting at my solves. Haha

[@Nevir202]

It's too late for shouting here, but in my head, there was much bellowing lol

[@aidentetreault3809]

😊😊😊😊

[@series161]

1:22:13 Simon: "Now are these resolved somehow?" - Me: "Yes" - Simon: "No" - Me: "Yes" - Simon: "No. Why can't it be a 5?" - Me: "Disjoint" - Simon: "I'm not going to be able to disproof that" - Me: "Why so stubborn today, Simon?"

[@thurst3224]

Series....Lol... "Why so stubborn...." Love it!

[@Wakaflockabach]

That meme from once upon a time in Hollywood where Leo is pointing at the tv... that's me every time I watch these videos, except I wouldn't have been able to start box 1 let alone finish it and I'm out here yelling about the simple shit he somehow always overlooks lol

[@MattiasDooreman]

Well, I figured that the digits refer to themselves in groups of 3, but if 5 would reference itself, you'd need 2 other such self-reference digits, of which there are none. But disjoint is the way.

[@Jeff_Saunders]

Why does he never listen to us while solving? It's very frustrating.

[@series161]

​@@thurst3224haha yes I have to laugh about it everytime I read it myself😂

[@victormanjarinsala2253]

At 33:28 Simon stops talking for a bit, looking very perplexed, and a cricket-like sound starts a couple of seconds later. Just perfect timing...

[@windybeach2184]

If we had a window view, we might have seen a tumbleweed rolling past

[@RMJ1984]

It's the CIA Signaling each other. he is under the supervisions!.

[@aawillma]

I think this puzzle is titled this because of a concept in cubing called commutators. In cubing, many of the algorithms used to solve the cube are built off of 3-cycle commutators. The full algorithm involves setup moves, the actual piece swap, and then the unwinding of the setup moves. Many of these "cycle" the location/orientation of 3 different pieces without affecting any other pieces. As far as I know, there are no algorithms that only affect 2 pieces on a cube, it will always affect 3 different pieces at minimum. Sometimes you have to do the process twice to get piece A to location C, etc. This self-referential rule creating a knock on cyclical pattern with 2 other spaces feels a bit like 3-cycle commutation moves.

[@victormanjarinsala2253]

That's my interpretation of the title as well :)

[@chipsounder4633]

Cool

[@yellingintothewind]

The Rubik's cube algorithm that averages the fewest moves needed per solve involves only 2 patterns. The first cycles 3 edge pieces (while rotating 4 corners in hard to predict ways, but you ignore them). The second rotates 3 corners sharing a face, with no other changes. The off-diagonal effects in this puzzle very _strongly_ visually resemble the pattern of rotations produced by that algorithm.

[@jkid1134]

Ah yes, a fellow PQP'Q' enjoyer. To your point about algorithms that affect two pieces on the cube: permutations work this way and must occur in pairs (if you say swap A with B and B with C as your pair, you get the mentioned cycling of three pieces), but you can change exactly the orientation of two pieces (if you have a cube on you, input [M'U]x4[UM']x4 and you'll see two edges flip).

[@chironchangnoi]

@@jkid1134long ago I had my solve time down to 3min and knew what the heck you were talking about. Then I wore out my cube and by the time I got a new one I had forgotten so much, and now I have forgotten everything. I ought to get back into it.

[@philipreinhold1180]

The fact that he had not one, not two but three disjoint deductions staring at him in box 8 for like 25 minutes was killing me.

[@talosedm8414]

I started skipping ahead cause it annoyed me so much lol

[@ezra7088]

Lol good find

[@OrionJA]

I let out the biggest sight of relief, around 47:00, when Simon *finally* used some number-cycle magic to eliminate that 1 in box 7 that was ruled out by the disjoint set rule 30 minutes ago.

[@MisterJDagger]

Thank you for your comment, that means my anxiety doesn't have to last for the next 15+ minutes.

[@nathanread7980]

damn you were not lying when you said disjoint subsets are your kryptonite

[@RedBarchetta2019]

This looks like the final boss of the 159 column trick. When Simon was solving box 1, it began to become apparent that there had to be a low digit, a middle digit, and a high digit in each row of a box. And sure enough it turned out that way for the columns as well. Fantastic puzzle!

[@colej.banning2419]

I was shouting "Entropy! It's all about entropy!"

[@Anne_Mahoney]

Yes, exactly: I noticed this very early on and was surprised it never occurred to Simon -- because I only got the idea from him in the first place! 😺

[@janerobson2297]

Same here. It was the only thing I deduced, and so didn't stand a chance of finishing the puzzle. I think Simon gets distracted by one thought and misses others. Not in this puzzle, but recently he's got hung up on parity and missed other more obvious or simpler deductions.

[@crazypantaloons]

​@@colej.banning2419 Yeah, both the XYZ rule and white dot set the entropy rules early. If Simon had spotted them, I suspect he would have solved this puzzle in half the time.

[@MrWaffles1030]

To summarize the interesting symmetries Simon saw. 1) In columns, the modular sets (147) (258) and (369) occur in the SAME box. 2) In rows the entropic sets (123) (456) and (789) occur in the same relative position in different boxes (in row 1, for instance, the low digits are in the top-right corners of each box, the middly digits in the top-left), and the high digits in the top center). And 3) along the negative diagonal, each digit appears precisely once. Beautiful stuff. There's gotta be some gorgeous math theorem underlying it all, but even absent that proof the results are a work of art.

[@crnmjgnthdrrgnl]

And in the positive diagonal the same 3 repeat (though I guess that's just combining other symmetries and sudoku rules, really, but I found it cute)

[@BryanLu0]

I think disjoint might be required? Within a box it is clear that each row and column must contain digits of different entropy, or else the digits will clash within a box

[@ianagol]

Yes, there is a kind of entropic condition going on in each box, each set of the digits 123, 456, and 789 occur in a cycle. This was helpful to me in solving the puzzle, since finding one cycle in a box (like the 123 in boxes 1,2,3,4,7) determines the other two cycles, which I colored. One can prove this cycle pattern for the off-diagonal boxes first, and then propagate it to the diagonal boxes. I didn’t notice these other patterns. It turns out that the cycle patterns in each box end up being the same pattern (maybe this holds in general).

[@cbates4545]

Theres also some magic square/15 sum thing going on in the columns. (1,5,9) (2,6,7) (3,4,8)

[@ianagol]

@@cbates4545 do you mean the rows in each box?

[@ravenous478]

It might take me an hour and a half to wrap my head around this concept.

[@veggiet2009]

Same, he keeps repeating these numbers and they kinda penetrate my brain

[@olivier2553]

I think that one way would be the internalize the reverse of the XYZ relation: as the 3 cells form a circle, it could help (make it easier to solve) to move around the circle in the other direction.

[@markjreed]

I think the XYZ formulation is a little abstract. Even a little punctuation would go a long way: if cell (X, Y) contains Z, then cell (Y, Z) must contain X. Applying that same rule again tells you that cell (Z, X) must also contain Y. If you try to apply the rule again you get back where you started. So every digit you place gives you two other digits.

[@windybeach2184]

I had to break out my notebook and colour code a little grid to follow the rule for the first hour. Then when it eventually became automatic to me, I only needed 15 minutes of sudoku to finish the puzzle 😅

[@ttbf]

Favourite line: 07:04 "They are going to be getting a.... an EMAIL after this."

[@inspiringsand123]

Rules: 01:42 Let's Get Cracking: 11:03 Simon's time: 1h18m25s Puzzle Solved: 1:29:28 What about this video's Top Tier Simarkisms?! Bobbins: 1x (23:21) Three In the Corner: 1x (56:47) Nori Nori: 1x (18:35) And how about this video's Simarkisms?! Ah: 20x (17:06, 24:14, 24:14, 24:37, 24:40, 27:55, 30:44, 34:36, 37:05, 38:50, 1:06:53, 1:09:01, 1:09:18, 1:09:18, 1:09:45, 1:11:24, 1:13:45, 1:18:15, 1:25:26, 1:28:03) Pencil Mark/mark: 10x (28:07, 45:59, 46:02, 55:21, 56:27, 56:29, 57:43, 57:56, 1:03:20, 1:05:52) Weird: 10x (15:35, 16:19, 48:46, 51:38, 51:49, 54:39, 55:04, 55:11, 1:00:50, 1:01:52) Hang On: 9x (04:32, 13:39, 27:48, 33:01, 52:23, 53:39, 1:05:42, 1:17:05) Sorry: 8x (14:30, 24:43, 31:03, 33:06, 37:55, 1:00:58, 1:28:14, 1:28:14) Wow: 6x (00:58, 00:58, 26:48, 1:10:16, 1:15:35, 1:29:29) Nonsense: 5x (23:21, 24:04, 1:11:14, 1:21:07, 1:30:30) Symmetry: 5x (34:39, 47:13, 48:15, 55:14, 57:40) What on Earth: 4x (03:32, 40:57, 49:56, 50:30) Bother: 4x (1:09:05, 1:12:06, 1:18:54, 1:21:04) Good Grief: 3x (47:38, 47:53, 54:03) Clever: 3x (41:46, 41:49, 47:53) Beautiful: 3x (47:38, 1:11:50, 1:17:05) Unbelievable: 3x (1:29:41, 1:29:44, 1:30:33) Progress: 3x (59:47, 1:07:02, 1:07:02) Goodness: 2x (58:35, 1:06:48) Lovely: 2x (56:45, 56:47) Extraordinary: 2x (09:37, 09:59) Ridiculous: 2x (06:33, 06:35) Surely: 2x (05:50, 19:29) Baffling: 2x (1:06:46, 1:13:16) Nature: 2x (34:14, 36:45) Cake!: 2x (09:21, 09:29) What a Puzzle: 1x (1:29:19) In the Spotlight: 1x (56:50) I Have no Clue: 1x (58:49) Brilliant: 1x (08:18) Incredible: 1x (1:29:29) Disconcerting: 1x (51:49) By Sudoku: 1x (18:58) Shouting: 1x (1:28:14) I've Got It!: 1x (47:34) In Fact: 1x (01:10) Obviously: 1x (31:47) Whoopsie: 1x (1:19:25) We Can Do Better Than That: 1x (20:02) Pregnant pause: 1x (00:38) Phone is Buzzing: 1x (21:15) Fabulous: 1x (30:11) That's Huge: 1x (1:16:47) Unique: 1x (06:46) Most popular number(>9), digit and colour this video: Ten, Twenty (2 mentions) Two (217 mentions) Orange (10 mentions) Antithesis Battles: Low (8) - High (1) Even (4) - Odd (3) White (3) - Black (0) Row (287) - Column (248) FAQ: Q1: You missed something! A1: That could very well be the case! Human speech can be hard to understand for computers like me! Point out the ones that I missed and maybe I'll learn! Q2: Can you do this for another channel? A2: I've been thinking about that and wrote some code to make that possible. Let me know which channel you think would be a good fit!

[@MarushiaDark316]

SPOILERS: Something I noticed while Simon was still finishing Box 1 is that this seems like a natural extension of the 1-5-9 indexing ruleset, wherein you apply modularity in two-dimensions. By that I mean ... in 1-5-9 puzzles, take Column 1 for instance. In each box, you select one low, one middle, one high because otherwise you get a clash in some other box when you go to fill in its index. This is something Mark and Simon are already well-familiar with. In this puzzle, take a look at Box 1. No two cells in a row or column are selected from the same modular group. M-H-L L-M-H H-L-M The reason it's called Rubick's Cube would then be because you start with numbers listed in order from 1-9 in each row and column. Then, you're sliding rows and columns around to jumble them up to produce this interwoven modularity pattern throughout the entire grid. The disjoint and kropki are then used solely to determine uniqueness. From the rules, Simon already observed you're cycling XYZ, so you could keep going XYZ-XYZ-XYZ ... take any three numbers from that sequence and you get a coordinate and its fill. If you think of XYZ as low-middle-high, you cycle LMH-LMH-LMH, which should be the final pattern and that's indeed what we see in the rows. Interestingly, not with the columns in every case, which could be a result of the position of the kropki dot being horizontal.

[@studgerbil9081]

this had me pulling my hair out because, as you mention, Simon figured this out ages ago.

[@windybeach2184]

I felt like I knew a lot of the logic as distinct units but my brain struggled to tie them all together tightly. So I had to plod along applying the XYZ to the easiest looking cells, and filling the gaps with disjoint and sudoku.

[@kathyjohnson2043]

Please, please, please Simon, never think that you are not clever or worthy of praise! You are amazingly good at what you do. We only shout because we care about you enough to want to help when you miss a trick!

[@ghandigoots]

I know when I shout, it's because he's taught me so well and I just want to help him find his mistakes

[@windybeach2184]

I return here because Simon is an amazing solver and an even better teacher! I feel smugly proud when I spot something before him 😂

[@Erikmitk]

Disjoints are truly Simon’s kryptonite. He wasn’t kidding! I was on the edge of my seat with some (to me obvious) pencil marks that could be deleted immediately! 😂 Or using the disjoint at the end to place digits instead of regular row/column sudoku!

[@katiekawaii]

Like at 53:45 when he notices he can't have a 1 in the 4th cell of a box and then immediately puts a pencil mark in cell 4 of box 8. 😅 I actually think it's really fascinating to see how differently different brains work. He sees things I would never spot in a million years, but at the same time, there are some things I find glaringly obvious that he can't spot himself. I think watching how he works has made me a better solver.

[@maljamin]

@@katiekawaii Exactly, he did that more than once. Disjoint drives sudoku in new box, same digit, same position, disjoint ignored! (But I'm here because my brain failed me i countless other ways and I kept breaking the puzzle)

[@jothki]

Whenever he finds a chain of forced digits he'll focus on that chain rather than on the consequences of each of the individual digits, only going back to identify other new restrictions based on a quick visual sweep. Disjoint sets are easy to follow up on if you actually pay attention to them when you mark digits, but it's even more resistant to that sort of visual sweep than the rows and columns are. In general, he seems to focus on individual critical points while paying less attention to contradictions outside of those points. It's probably helpful for the advanced logic that he needs to do, but it does make him a bit sloppy when it comes to identifying more basic logic.

[@windybeach2184]

@@jothkiI think you’re onto something, and it’s what makes him amazing at breaking in seemingly impossible puzzles. He’s able to focus on a challenging concept without being distracted by the smaller details that pop up along the way. Being great at scanning digits is handy in normal sudoku puzzles but these variant rules require a lot of extra thinking.

[@youresogolden_]

As someone who is relatively new to the channel and sudokus with any other rule sets than normal sudoku rules (heck, four weeks ago I didn’t even know there were sudokus with other than normal sudoku rules), I can’t even fathom how someone could solve this. I’m amazed every time by Simon’s ability to crack them (even if it takes him a bit, idc because I always enjoy the ride and learn a lot!)

[@ghandigoots]

I've been watching for a few years and still have no idea on where to start with some of these puzzles. When I first heard about people shouting at him, I thought it was utter nonsense. He does miss some easy things here and there where I get to shout at him, and one day, you'll probably join those ranks too because he's so great at explaining every logical step he takes. I really love watching him open puzzles live and especially this one. Watching his brain work to understand the rules and play with them at the beginning to figure out how to crack it was awesome

[@windybeach2184]

He’s a fantastic teacher! Stick with it and you’ll be amazed at how the concepts start to come together for you. It’s always tricky when learning a new rule set but a lot of them become familiar quickly.

[@thejuggler42]

Thanks to Simon for tackling this absurd multi-dimensional crystal of a sudoku--even if it wasn't entirely voluntary! :) It's not an easy puzzle, so going in blind and cracking it in an hour is certainly nothing to sneeze at, and the break-in especially was a joy to watch. There's a LOT to say about the math here (too much for a KZbin comment), but I will say that if you treat each digit and row/column index as 0-8 instead of 1-9, and then write them as two-digit numbers in base 3 instead of base 10, the relationship between entropy and sudoku boxes and the relationship between mod-3 and disjoint groups might become more apparent.

[@MaxHaydenChiz]

This is brilliant. And your base 3 explanation is really helps understand why this works. What inspired you to come up with this?

[@thejuggler42]

I was inspired by conversations with my friend Ash (who has a PhD in mathematics). They noticed that in Latin Squares there's no functional difference between row, column, and digit, but were annoyed that the same symmetry is not true of Sudokus because of the Sudoku boxes. Our goal was to fix Sudoku so that it regains this hyper-symmetrical nature, and I found that it can be done with the XYZ rule, which forces the sudoku boxes to be "rotated" from the row/column space into the digit/row and digit/column space. ​@@MaxHaydenChiz

[@robinbrown6530]

Maybe see if Simon would want to do a video with you explaining this in detail? Or maybe Numberphile would pick it up. IDK, I just think it would be fascinating to see how this whole thing ticks behind the scenes.

[@thejuggler42]

@puzzlepusher is about to put up just such a video on KZbin. Stay tuned :) @@robinbrown6530

[@puzzlepusher]

@@thejuggler42*ahem* What happens when you project a Rubik's Cube onto a Sudoku: A lesson from the setter! kzbin.info/www/bejne/nZDRi2l7aNiahLM 😊

[@OlafDoschke]

18:15 pure comedy, I love it, Simon.

[@MrTyler918273]

reminded me of the Knights who say Ni saying "it".

[@Lucashallal]

I laughed my ass out seeing that

[@landonmenne]

When I realized how the numbers rotate through the puzzle the Rubic cube reference made a ton of sense. Thus puzzle is so creative. One of my new favorites.

[@Crystal-dc3oy]

This puzzle has captured my mind like no other CTC video I've watched. I stayed up late trying to work out the underlying logic. It felt like being back in my college years, in my favorite class of my whole math program. I am incredibly grateful to you and the puzzle creator for reigniting this excitement. I've made some progress, I think, and I might share it later when I have a more complete picture, but for bow, I'll just say that I found the notation "rXcYdZ maps to rYcZdX", where d is the digit that placed in the cell, much easier to understand.

[@thejuggler42]

If you're interested, I explained what I understand about the math behind this puzzle in a video on PuzzlePusher's channel (also titled Rubik's Cube) :)

[@bsketbllplyerable]

Boxes 1 5 and 9 are interesting here because they have values that "self-index". As you saw with box 1, values 1 2 and 3 are. Constrained. As you found: 1 can only go in 1,1 2,3 3,2 2 can only go in 2,2 1,3 3,1 3 can only go in 3,3 1,2 2,1 Similarly for box 5 4 can go in 4,4 5,6 6,5 5 can go 5,5 4,6 6,4 6 can go 6,6 4,5 5,4 Box 9 follows

[@Rubrickety]

Solving this made me deeply appreciate how badly suited human brains (mine, at least) are for certain tasks. It felt like trying to paint a picture with a hammer.

[@Zooobly]

oh man, hats off to Simon, I could NOT have done this, NO WAY I would have gotten that break-in. But oof, that disjoint set really gave you trouble, haha. 1, 2 and 3 in box 8 being available from 56:47 (from box 3) made me despair, just a little. At least you did end up spotting it about 15 minutes later

[@LouisBarjon]

I screamed so hard in my head about this, especially because it gives the position of the 3 in box 8

[@gamerbb2]

I liked the indexing of this puzzle and how there was a global relation between modular numbers in the columns and the spacing of low, medium, and high numbers, which Simon pointed out.

[@margaretsinclair6697]

45:29 looks like this will be not just disjoint puzzle, but also like a 159 puzzle, there has to be a low, middle and high digit consideration so that we don’t get a repeated digit in a box from the XYZ rule. 159 feels 2 dimensional. This XYZ feels 3 dimensional - like a cube.

[@FluffSWE]

i have done the same thing as Lucy. but for me it was a kidney :) 12 years ago this year and havent regretted it one second. but i fully understand those that see it as to much, but for me it was same reason as Lucy. right thing to do.

[@d_mosimann]

This puzzle CLEARLY has something to do with Rubik's Cube! - This puzzle is all about sequences, just like solving a Rubik's Cube is all about algorythms - which is the same. - To mentally switch numbers XYZ into YZX, is like imagining to switch 3dimensional cube parts. - 3 cells in this puzzle always relate to each other, just like 3 cube parts always relate to each other in Rubik Cube's algorithms, too. - The tripplet patterns in this puzzle are all over the place, exactly like in the Rubik Cube's algorithms, too. - The two top cells in this puzzle have additional rules which make it easier, just like for solving the top layer of a Rubik's Cube - one does not use the standard algorithms there but easier ones. - and the list goes on

[@TurquoizeGoldscraper]

I got the concept and figured out several numbers, but I started making mistakes and gave up. Good job Simon on keeping the numbers straight.

[@humanbirdsong]

Fascinating puzzle. Within every 3x3 box, lows (123), mids (456) and highs (789) never repeat in a row or column.

[@karentighe8940]

I DID IT!!! I can’t believe I actually did this with no hints. Thank you CTC for improving my puzzle skills!!!

[@MarkBennet10001]

Love "nori nori in reverse" - just what I needed after an intense few days without puzzles

[@stevieinselby]

Nori nori in reverse - that would be iron-y 😎

[@fulltimeslackerii8229]

So iron iron?

[@longwaytotipperary]

@@stevieinselby😄

[@chocolateboy300]

I got 236 minutes. I was lost at the beginning and couldn't proceed, so I decided to give up and watch the video only for Simon to remind me of the disjointed rule. After that, it was still incredibly difficult. It is very hard to scan and keep this geometry in my head. I wasn't even planning to do this puzzle, but I made early progress by noticing that each box 1, 5, and 9 had to have an all or nothing with cells that reference themselves. I gave it a go and I'm glad I did. Very hard, but fun!

[@koocnaed]

The fact that I managed to solve that in under an hour, without following along with the video makes me very proud. That is going in my record of achievements folder!

[@JorgePacker]

The modularity thing did help me with the negative diagonal! I really enjoyed this sudoku!! What a brain twister!!!

[@rossrodrigue1936]

A puzzle specifically designed to make Simon practice, his disjoint scanning. All jokes aside though, quite an interesting puzzle, the way, the numbers interplay with the geometry of the sudoku grid

[@laurasmith2173]

I've been waiting for a puzzle that Simon opened live on screen and I'm so happy to see one. :) Not to mention it's disjoint!

[@SirBradiator]

When you said Lucy was hoping for some long videos for her post op recovery, I looked at the video length and was debating whether 90 minutes is considered "Long" for your channel. most KZbinrs this would be considered very long, but by your recent standards this is probably about average. 😂 Best Wishes to Lucy.

[@mstmar]

i did find one thing underneath everything that isn't modular nor what was going on in the rows that i can actually explain. what i found was entropy. Simon did find something similar to that with the white dot, but it extends to the whole grid. i'll start with an example, if you look at the first row of box 4. this has 41a, 42b and 43c, where a,b and c are the digits in those cells. by the xyz rule, we need to put a a 4 in r1ca, r2cb and in r3cc. we know that one of those is in box 1 (i.e cols1-3), one in box 2 (i.e. cols4-6) and one in box 3 (i.e. cols7-9). so a, b and c have to be one low, one mid and one high. you can repeat the same reasoning for any row of any box, or any column of any box as well. this means that every row and every column of every box is entropic with this, whittling down some of the possible values was much quicker. at 55:30 for example, we have a 1 in the middle column of box 5, so there can't be a 2 in r6c5, which then takes out the 3 from r4c6 without having to think about the xyz rule again.

[@thejuggler42]

Well spotted!

[@Arcessitor]

God I was looking for that line of reasoning for so long but couldn't find it. Thanks a lot for clarifying this for me.

[@longwaytotipperary]

I’m amazed at how Simon can hold so much information in his brain! I had to write the rules down and constantly look at it to even follow. Great puzzle, great solve!

[@yellingintothewind]

You weren't wrong when you said Disjoints are your bane. Starting at 48:30, you could place a 3 in box 7 by disjoint, and by the hour mark you have 4 digits available by disjoint.

[@angec9908]

Michael sure is a prolific setter. I think I’ve watched 3 or 4 videos of his puzzles going back a couple of months. The first one Simon corrected his pronunciation of Michael’s name. So I guess this is the one where he got it wrong. And here I thought Simon was perfect :)

[@yellingintothewind]

At 1:25:00 you can disprove the 5 in the middle, because there are 81 (0 mod 3) cells total. Each cell either orangifies 1 or 3 cells, so if the middle is 5, you need two other entries that self-index along the diagonal, but disjoint prevents that immediately, so _no_ digit along the diagononal self-indexes. Of course, you can also disprove it and the 4 by disjoint directly....

[@robinbrown6530]

So I solved this and it was super weird. I got to about where Simon was at 1:15:50 and could not for the life of me see that disjoint trick with the 9s. But I did see that the grid had a Mod 3 pattern in the columns and a Low, Middle, High pattern in the rows. It wasn't that I just saw the pattern and was like "If I follow the pattern maybe it solves." I KNEW it would solve off that pattern. And I KNOW there is SOME mathematical logic going on that proves it. I just have no idea what it is. Its the first time I have ever solved a puzzle knowing I am using a logic I do not understand and cannot explain. Its a good thing Simon found that disjoint trick with the 9s cause "This is logical, just trust me." does not make for a reassuring rational for a solve. I am convinced as well that if you do know the math behind this, what ever it is, you could probably fill this in very quickly. like sub 10 minutes if your also fast at Sudoku. For me, it took about half a day staring at this thing on and off again throughout. I am going to go to bed now Michael. My head hurts. Thank you for that.

[@nedb147]

I’m so surprised that Simon didn’t also notice the roping in the rows as well as the columns. A man with such a brain for numbers would have found that very satisfying

[@sir3482]

there was an awesome entropy discovery in the rows made so elequontly by the kropki dot. Simon began to use it for the 67 pair on the dot but never fully realized it even when working out that 45 pair in the same box. It wouldn't have taken him so long had he noticed it and used it.

[@Salman-os7pr]

As a speedcuber and sudoku lover, I am excited for this solve! But ofcourse i am not going to attempt a solve that takes Simon over an hour!

[@kristerlarsson4633]

You should try it. It's by no means easy, and the start is really hard. But Simon got stuck several times by not thinking about the rules, at least 30 minutes went to that. In addition to that the rules makes it take time to find places for digits without it being hard.

[@chipsounder4633]

That was my new favourite.. the thing i noticed quite early is the rows and columns were all roped as well as the disjointed triple sets ❤❤❤ epic

[@Al-lh2oo]

Simon, I think your end comments are very relevant. There seems to be an interplay of some sort between the indexing and the disjoint that needs further investigation. It puts a negative constraint on some cells that is not easy to spot. Mostly, I think, because we are not trained to look for it. Great video btw. I watch each day from Geelong. You can look up where that is.

[@emilywilliams3237]

Fascinating. Both from the puzzle's perspective (the design and "discovery" factor) and the commentary. I think that one of the reasons this was the kind of video that it was is that indexing is a bit too administrative to suit Simon's natural bent, and does not in itself have enough of the wider-ranging logical elegance that he loves. I will definitely try this puzzle at some point, though, because administrivia is right up my alley. Thank you, Simon, for the solve - I always love your videos!

[@shotdoctor5869]

I feel like row should be Y and X be column. You move horizontally along the X-axis and vertically on the Y-axis.

[@uigrad]

I loved this so much! I finished it faster than Simon (54:35 for me), but I'm still don't feel like I fully grasped it. It's so cool how one tiny deduction at the beginning just keeps going and going and going!

[@Qurqirish_Dragon]

39:45 Haven't watched the video yet, but after about 10 minutes of "what does this do?" I realized that every row/column within a box had to have one low, one medium, and one high digit, and then it was simply a matter of determining one digit in a box to get all the appearances of its tier (low/medium/high) in 3 boxes, with the exception of the 456 in box 5. Curious to see how Simon does this, since once I saw that, it went very fast.

[@sacredsock8031]

I feel that at 25:00 there is some sort of generalisation that can be made involving the word entropy

[@everorizon]

I've noticed a couple of things in this puzzle. - In any box every stripe of cells must contain digits from different entropy sets because each cell will index the same digit, either in consecutive rows or consecutive columns. For example, r1c123 will index 1 in rows 1, 2 and 3 respectively. If two of them was from the same entropy set, they would index two 1's in the same box. - In any row or column, each set of cells in the same relative position of their box contain digits from different modularity (mod 3). Otherwise it would index a digit in the same relative position in multiple boxes. For example, if r5c14 was a 36 pair they would index two 5's in the first cell of two different boxes (either boxes 1 and 5 or boxes 2 and 4). The modular thing might be a coincidence though. Or it might not because of the disjoint set rule?

[@KrisCadwell]

Bravo Simon for solving this. I don't have the patience or the working memory to even attempt it.

[@SamuelSandeen]

I noticed something beautiful at 1:23:01. The only digits that can point to themselves are on the diagonal and 6 digits are already on the diagonal so for any digit to point to itself on the diagonal all three of the remaining digits would need to since the the number of digits pointing to themselves need to be a multiple of 3. And 4 can't point to itself by disjoint therefore r5c5 isn't 5. I didn't use this in my own solve but it's quite pretty.

[@MaierFlorian]

Seing the solution of the puzzle I am certain that there is a mathematical principle you could prove at the start of the solve, allowing you to like speedrun the puzzle in a few minutes - unfortunately I don't have a clue what this principle may look like :'D

[@HunterJE]

Yeah so much weird emergent stuff here feels like you could write whole papers about this puzzle

[@shoezomaku]

In case you're wondering (like I was), yes, he does see the disjointed numbers in box 8 at 1:09:50.

[@stangerrits6712]

A ‘disjoint selection tool’ could help a lot in solving these puzzles. You should somehow be able to select a disjoint set, e.g. if you double click on R1C1, all the top left cells in all other boxes are selected. I think this will make scanning much easier.

[@benwhite5734]

Simon's face when trying to get his head around the rule summed up my feelings perfectly. By the way in addition to vertical roping there was horizontal roping and patterns along all the diagonals, as is common in miracle puzzles. As Simon noted, it's too bad you can't just assume the patterns and fill in all the digits.

[@piarittersporn]

What a massive puzzle. Without paper and pen for notes, my brain would have been in pure chaos and wouldn't have been able to solve the puzzle in several hours. But that's how I was able to do it.

[@wilesfiles1]

It took me 128 minutes to solve. It would've taken much longer if I hadn't noticed the modularity of the columns. It was a beautiful puzzle. I think starting in the upper left corner with the white dot was the way to go.

[@aaronsebastian5156]

Fun puzzle, glad I wasn't the only one struggling to grasp the XYZ concept. Simon noticed the 147, 258, 369 in the columns of each box, but didn't notice the 267, 348, 159 across in each box. There was a distinct symmetry within the boxes, which is difficult to explain until the pattern emerges, that once you could see the first couple of boxes getting filled, the rest all followed the same pattern. I thought the diagonal was actually easy to rule out 111, 222, etc. because the math wouldn't have added up. Triples, like 111, blocks off only 1 coordinate, where as 112 would technically block 3 coordinates (even though that doesn't work for the solution). With the XYZ rule, each coordinate set and number creates a trio, so everything operates in groups of 3. Once it was confirmed 1,2,3 couldn't go in the first three locations of the left to right diagonal, there wasn't any possibility of any of the other 6 digits to go in those slots because of the trio nature of the XYZ rule. There would've needed to be at least one group of three in that diagonal that would also follow a LMH pattern to fit the solution (159, 147, 258, etc). Ruling out the low three disproved the middle and high sets from using that pattern, as there would be no low numbers to counterbalance the middle and high sets.

[@psiphiorg]

What a wild ruleset! Discovering the entropy and modularity tricks that define the rule, as well as how the disjoint subsets amplify that, took a while to wrap my brain around. In the end, both my time and solver number ended up being just under a milestone: 7 seconds under an hour (59:53), and 2 solvers under a thousand (number 998).

[@NicD]

54:08 hurts, looking at one disjoint for 1 and missing the other... Edit: 1:09:19 was cathartic!

[@DarkChasm]

I haven't watched the video beyond the intro, but a) I'm in love with these mad formats and b) seeing Simon reaction is priceless :) This reminds me so much of the mad miracle/friendly indexing miracle by thoughtbyte :) Go watch that if you haven't to see Simon go loopy :)

[@karsaanita]

What. A. Puzzle! Best two hours of my day today.

[@Yalsof]

I love Simon's breakthrough for the first digits. It's much more elegant than my brute force method of the white dot. I focused on r2c1 and it's relation to r1c1 and it turns out there's only one valid option for that domino.

[@imatrot]

If Simon was one of my Honours candidates, I would be commenting that his thesis shows great insights that move the problem forward but sometimes he doesnt follow through with the gained logic and thus leaves easy conclusions unanswered.😂

[@anaayoung9142]

If Simon says: it is very hard to understand... imagine me?! I was only going by the flow.... aham aham, Yes, that's it..😅 Didn't thinking at all about XYZ or YZX.... super confusing for me. I am glad you can solve this type of puzzle for us! Thank you for the amazing solve 😀 Ps: when Simon need to do a puzzle faster he relies on sudoku 🤔😂

[@yannayli]

The Rubik's cube reference is quite clear: it's a huge 9x9x9 cube where the x-y axes are the gird and the z axis is the value in the cell. The XYZ restriction becomes some sort of algebraic symmetry group and I haven't figured out the rest (linear transformations?). I will think about it when I take a shower today :) My guess is that the disjoint restriction reduces the possible solutions to 3 and the dot takes it down to one. But I'm not sure.

[@thejuggler42]

That's a very good way of thinking about it! Without the dot, there are 12 solutions - 4 where the long diagonal is all self indexing, and 8 where it's not at all self indexing.

[@yannayli]

@@thejuggler42 Thanks! Is that how you set it up? First analyzing the algebra under the xyz rotation and then added the dot to enforce uniqueness?

[@thejuggler42]

Yup, that's the short version :) @@yannayli

[@yannayli]

I envy your ability to reason about this six dimension modular field. I had my shower and wow this gets complicated. I figured out some of it, but I need to develop the algebra/geometry further. I saw you said you are going to make an explainer, I'm looking forward for that :) Does the white dot have an algebraic Interpretation as well or is it just a handy way to trim down the options?

[@thejuggler42]

@@yannayli You're in luck, @puzzlepusher just posted a KZbin video where we talked about some of the math behind this!

[@TeaAndTea96]

The amount of collars is ever-increasing!

[@marinerabanal7599]

I'm very proud to say I solved it in 88:33. The hardest part was the XYZ rule, I had to say the digits out loud to avoid mistakes. And to think that when I started watching this channel every puzzle seemed impossible and now I try them and from time to time I can solve them. It makes me happy ❤

[@Kirbyfan87827]

Finished in 32:29 by following along with the video.

[@rainerzufall42]

Very clever rule set! Awesome!

[@Slimeyen]

The comment at 18.15 may be one of the funniest things I've heard in a long time :D

[@eve_the_eevee_rh]

37:45 that was really fun lmao! I love these unique rulesets where the puzzle is more of finding interactions in the rules than the classic rulesets- another puzzle I remember this for was something related to mod 3 and low/med/high digits in the rows and columns- don't remember which one that was, it was a while ago, but that one was fun as well! Great job Michael!

[@BeachHutStuff]

I’m sat watching this on a train, and next to me is a man doing a Rubik’s Cube 😂

[@AugustoValentini]

Now we need a Numberphile video explaining why this rule set creates the modular pattern in the columns and the entropic pattern in the rows

[@thejuggler42]

I'd love to see that! ;)

[@jimi02468]

That must happen.

[@basilefff]

This was an interesting one. I should have seen symmetry between 1-3 and 8-9 earlier, but 01:39:10 is a fine result

[@MrMouski]

Would be nice to add a disjoint button on Sven's Sudoku Pad. It would look like a box and when you click on a particular square, all 9 of them would light up in the grid.

[@thaumaTurtles]

I'm surprised no one has pointed out that the rows within each box _also_ come in sets, just as the columns do. Each row contains some cycle of the numbers 4 8 3, 1 5 9, or 7 2 6. Interestingly, you can think of each number in these sets as being 4 more than the previous one, if you count the numbers in a cycle. This does break for 3 4 and 6 7 though. Absolutely fascinating construction

[@r0bbiegill]

I took a bit more of a brute force approach. I very quickly understood what the white dot was doing, but then needed to do lots of pencil marking about what possibilities existed. While there was a lot of back and forth, I really appreciated the almost algorithmic way deductions flowed.

[@artyonehundred]

Box 8 gets no love and there is a 1 and a 3 that can be eliminated with the disjoint condition! (I'll stop shouting at the monitor now!) I never cease to be amazed at how Simon sees really hard things and then misses the easy stuff. Not that I could solve this in a month of Sundays. Wow!

[@vladyslavvlasenko9161]

Well, he did warn that disjoint was his kryptonite)) Otherwise - brilliant solve!

[@HunterJE]

24:00 Interesting to note that the pattern of pencil marks here takes the general shape of the fill on a 3x3 latin square...

[@HunterJE]

So much weird emergent stuff arising from this ruleset; entropic alternation, mod-3 sets...

[@rainerzufall42]

58:37 YES YES YES, 3 can't be top right!!! (1 can't either, but that was longer ago) Eliminate it, and set r9c6 to 3!

[@andreaslooze]

Shouting at the screen :)

[@rainerzufall42]

@@andreaslooze LOL, yes, exactly!

[@longwaytotipperary]

❤ Amy’s poem! Lucy is a saint! ❤ the puzzle !

[@AnaFromTheShire]

Awww thanks :D Didn't expect him to read it, was a lovely surprise

[@longwaytotipperary]

@@AnaFromTheShirelovely for us to hear also!!

[@thejuggler42]

There is so much to love about this poem. Are you planning to publish it? "Cuts time" and "rattling flax" are such beautiful ways to describe the ticking of a second hand, and the latter is all the more apropos because flax is used for weaving rope, and because the flax plant (at least in Polynesian culture) is a symbol of the different generations of a family. @mTheShire

[@AnaFromTheShire]

@@thejuggler42 Gosh, thanks! Oh haha no, no plans to publish, I'm not a writer or a poet, it's just a little thing that came out one time and I thought might be up Simon's alley lol Yes, flax (harakeke in NZ) is also quite important in te Ao Maori.

[@longwaytotipperary]

@@thejuggler42 your puzzle is spectacular! Taking time to give positive feedback to Amy doubles my appreciation of you!

[@billyoung8118]

I have a degree in electrical engineering, and am a statistician for a living. I can't "math" these rules! So few words but so hard to grasp.

[@Raven-Creations]

That was a head scrambler, at least at the start. My first deduction was that the dot had to be 34 or 67, because the digits had to straddle a box boundary, because one places a 1 in R1 and the other places a 1 in R2. 34 can be ruled out because 3 can't go in the corner, and 4 in the corner would break the disjoint set. This meant it was 67, and I also shaded cycles, but there's only need for one colour. A cycle is either complete or it isn't. If you leave un-processed digits white, you can instantly see that they need to be processed, rather than having to hunt for stray instances of a colour. @ 24:21 - "This is going to go 1, 2, 3, 4, ..." - No it's not, because that 1 in the corner can't go next to a 2. There are only two places for 1, and whichever it is creates a 123 cycle in box 1, so 2 can only be in R1C3 or R3C1, and 3 can only be in R1C2 or R2C1. @ 30.34 - "The problem is this isn't under any pressure" - Oh, but it is, it has to be 67, and has to be in that order because 7 in R1C1 would break the disjoint set rule. @54:17 - "Can we keep that going?" - Yes, but your brain won't let you stick with what you were working on and makes you try something else. You just ruled out 1 from position 4 in box 5, so why not rule it out from position 4 in box 8 leaving just one place for it, in position 1? Your brain is kitten-like, once it's caught the string, it looks for something else. You need to develop a terrier instinct, where you keep on at something until it's dead. @ 1:25:53 - "So let's try 4 there" - 4 is already placed in position 1 in box 3, it has to be 9. Part of my methodology was to take say the 2s pencil-marked in box 5, that is R5/6C6=2, so R6C2=5/6, and R2C5/6=6. This provided useful pencil-marks. After checking the disjoint sets, usually something could be eliminated, leading to three more digits, which further impacted the sets or allowed more sudoku. Eventually, it reached critical mass, and everything collapsed. In terms of analysis, the rows are a bit like 159 puzzles, in that they have to have a low, mid, and high in each box, because the numbers in C1-3 place the row number in R1-3, C4-6 place the row number into R4-6. etc. I couldn't work out whether they have to be in the same relative position, or whether that was just a facet of this puzzle. It's not just modular roping in the columns, it's identical roping (i.e. the sequence of the whole column is just shifted by three and six and wrapped in each group of three columns.

[@AllenMS828]

I'm not often able to solve a puzzle where the video is over an hour and a half long, but I got this one in just under an hour and a half myself. The big difference between my solve and Simon's is that I actually like disjoint subset puzzles, so that helped level the playing field between our times. 😆 I started with the Kropki dot and basically tried every possibility for cell 1,1 until only one remained viable. From there, I looked at where 1s were still possible in the other boxes, then 2s, then 3s. I ended up with a weird half-solve, where all of the first three rows and first three columns were completely filled in, along with all of the 1s, 2s, and 3s in the other four boxes. From there, it was mostly just normal Sudoku, with only a couple places where I had to use the XYZ or disjoint rules. It was a very fun and interesting puzzle!

[@seeker606]

Loved the logic in this puzzle

[@richardstevenson371]

1:09:15 Simon finally sees it! That was driving me insane for 20 minutes! Love you Simon ❤

[@LarkyLuna]

XYZ -> YZX -> ZXY -> XYZ It's a left rotation if you imagine the three digits are in a cycle Which means that row X indexes column X and column Y indexes row Y There are so many ways to see this puzzle, very fun

[@seanfarrow9417]

I started in 1:1 and 1:2 once I realized it was going to put a pair of ones diagonally next to each other and the only possibilities are 3/4 and 6/7 and the 3 doesn't work because it puts two 1s in the first box. That helps unlock it a bit because then it puts a bunch of pressure on the 1 in box one and then that unlocks more numbers. If you're given a domino, use the domino. :) From there it's just kind of brute force logic on the possibilities

[@Raven-Creations]

That's how I started too. I couldn't understand why Simon ignored the dot at the start when it's the only clue we're given.

[@darci4434]

I'll attempt this puzzle until the end no matter what cos I'm massive cuber. I own 42 different rebids cubes. I'm not saying I'll solve it but I'll try to do it, I've solved some of your sudoku before but I will do that for sure no matter how long it took for everyone, looking forward to do it

[@thejuggler42]

Best of luck! I will say that while the name fits for thematic reasons (there are lots of 3x3x3 symmetries, like a cube, as well as a "rotation" rule), the techniques for solving it are pretty much unrelated to how you'd solve a Rubik's Cube.

[@darci4434]

yeah I understand why this sudoku called Rubik's cube because of rotations and it make sense because same thing to Rubik's cube the rotations will always come back too same colour so in sudoku its a cell. I'm currently doing this sudoku and I'm doing well so far. 1 hour in and got 8 digits but 9 pairs of xyz @thejuggler42

[@darci4434]

Not trying to be cocky but I expect that I won't achieve it cos I'm stuck at some moments but I'm not joking I just finally solved it cos I didn't give up@@thejuggler42