The Modular Miracle Sudoku

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** TODAY'S PUZZLE **
This was recommended to us by Clover, who tested it recently. It's called Cross About Dominoes and its the work of the math professors FullDeck and Missing A Few Cards. Oh... and it's absolutely extraordinary.
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Rules:
Normal sudoku rules apply. In cages, digits must sum to the small clue in the top left corner of the cage. Every diagonal of length 7,8 or 9 is either modular or unimodular. On modular lines, every set of three sequential digits contains one digit from {147}, one from {258} and one from {369}. On unimodular lines, all digits are from the same class (ie all from {147} or all from {258} or all from {369}).
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[@HunterJE]

17:50 3 in the very middle of the grid, right at center stage - you could almost say that's 3 in the spotlight...

[@LeDoctorBones]

And there were a whole two 3s in the corners too.

[@michaels4340]

Trying to keep up with you and your logic, but I don't know if I can do it... Oh no, I've said too much! :X

[@BeheadedKamikaze]

@@michaels4340 You haven't said enough! I thought that I heard you pencil marking. I thought I heard Phistomephel's Ring. I think I thought I saw you reach 5 (hundred thousand subscribers)!

[@dubtak4976]

Omg. EVERY positive diagonal is modular and EVERY negative diagonal is unimodular. That is just showing off by the setters, and I love it.

[@missingdeck9999]

[@WayneDupree1101]

So... got hooked on to this channel a few months ago thanks to my oldest child who started watching because they wanted to learn more sudoku techniques. I picked up many of the "Simarkisms" at first then got it to watching with them and now we watch every day. While we were watching one day my oldest said "I wonder how many times Simon's said 'bobbins'?" - well I came across Inspiring Sand's excellent comments with all of that information broken down and decided that I may not a great sudoku solver - but I am a developer so why not use the KZbin API and parse all of Inspiring Sand's comments and add them up... so, I did. In appreciation of his 800th video comment (now actually up to 803), Mark and Simon's 5 year anniversary and reaching the 500,000 subscriber milestone (including us now) I give you: Cracking the Cryptic 'Simarkism' Statistics First video with comment: 4/12/2020 Last video with comment: 8/31/2022 Number of videos with comment: 803 of 2897 (27.7%) They've said "Ah" 4348 times, an average of 5.4 times per video. They've said "Sorry" 3416 times, an average of 4.3 times per video. They've said "By Sudoku" 2656 times, an average of 3.3 times per video. They've said "Hang On" 2463 times, an average of 3.1 times per video. They've said "Beautiful" 2237 times, an average of 2.8 times per video. They've said "In Fact" 2165 times, an average of 2.7 times per video. They've said "Clever" 1757 times, an average of 2.2 times per video. They've said "Obviously" 1632 times, an average of 2.0 times per video. They've said "Snake" 1533 times, an average of 1.9 times per video. They've said "Lovely" 1218 times, an average of 1.5 times per video. That's the top 10, other notable phrases: They've said "Bobbins" 466 times, an average of .6 times per video. They've said "Brilliant" 1067 times, an average of 1.3 times per video. They've said "Good Grief" 1063 times, an average of 1.3 times per video. They've said "Maverick" 185 times, an average of .2 times per video. They've said "Pencil Mark/mark" 790 times, an average of 1.0 times per video. They've said "Phistomefel" 610 times, an average of .8 times per video. They've said "The Secret" 482 times, an average of .6 times per video. They've said "Three in the Corner" 154 times, an average of .2 times per video. I'm also working on total time solving puzzles, and some other stats so - like Inspiring Sand - if you have any suggestions on what you'd want to see based on his data, let me know. Thanks again Mark and Simon, keep cracking!

[@esajpsasipes2822]

i wanna see three things: - how many times has each number/digit/colour won in the "most popular" section - how many times has each participant in the antithesis battles won - an ultimate antithesis battle - add up all the mentions to see who wins in the respective pair

[@emilywilliams3237]

I LOVE this. I have loved Inspiring Sand's analysis, and this is tally is super!! I especially like "Good Grief" and "Hang On"!

[@WayneDupree1101]

@@esajpsasipes2822 Will look to add that info - I'll let you know what I find :) Thanks for the suggestions!

[@WayneDupree1101]

@@emilywilliams3237 Thanks Emily! Glad to know I'm not the only data geek out there! 😁

[@Paolo_De_Leva]

😂 Very cool❗ I didn't know the KZbin API was so powerful. I am glad the most repeated Simakism is "Ah." That's why we all love CTC.

[@missingdeck9999]

What a lovely solve!! Thank you so much!!! What a privilege it is to be featured!!! And thank you so much for the kind words!!!

[@kathyjohnson2043]

WONDERFUL! So much fun watching Simon's joy at discovering the pattern. Both the overall pattern and the carefully crafted killer cages that were the key to putting numbers in the pattern made this puzzle one of the most miraculous of them all.

[@BozoTheBear]

Such a brilliant puzzle, thanks a lot! (Cheers from an Aussie mathematician.)

[@bobblebardsley]

I love how the "every 3rd cell" logic acts almost like a knight's move constraint, I always feel like you need some kind of propagating constraint like that for a puzzle to collapse miracle-stylee once you've placed the first 2-3 digits.

[@Jigkuro]

"That's a three by green!" Not sure why but that line really got me. 😆

[@inspiringsand123]

Rules: 05:07 Let's Get Cracking: 08:04 Simon's time: 37m44s Puzzle Solved: 45:48 What about this video's Top Tier Simarkisms?! Bobbins: 1x (11:32) And how about this video's Simarkisms?! Hang On: 13x (08:39, 12:44, 17:08, 24:39, 25:02, 30:28, 30:28, 31:06, 31:06, 31:06, 35:30, 35:41, 35:53) Ah: 13x (13:21, 14:58, 18:52, 18:56, 27:37, 29:44, 30:28, 31:42, 32:32, 33:15, 33:17, 37:35, 39:51) By Sudoku: 8x (16:50, 37:24, 38:38, 40:03, 40:14, 40:18, 42:35, 45:01) Good Grief: 5x (27:43, 37:15, 43:03, 43:07, 44:02) Obviously: 4x (02:13, 07:47, 16:34, 20:15) Sorry: 3x (09:21, 26:42, 45:50) Clever: 3x (01:09, 32:39, 32:42) I Have no Clue: 3x (10:27, 13:44, 13:44) What on Earth: 2x (08:36, 34:10) Goodness: 2x (23:50, 27:39) Beautiful: 2x (23:50, 23:54) Magnificent: 2x (22:44, 22:48) Stunning: 2x (27:46, 27:46) Wow: 2x (14:58, 15:01) Useless: 1x (08:25) Apologies: 1x (40:58) Nonsense: 1x (29:53) Insane: 1x (44:41) Brilliant: 1x (02:18) Take a Bow: 1x (46:24) Shouting: 1x (41:00) Surely: 1x (22:39) I've Got It!: 1x (21:45) Unbelievable: 1x (38:48) What Does This Mean?: 1x (05:47) That's Huge: 1x (36:39) Pencil Mark/mark: 1x (42:40) Cake!: 1x (02:22) Symmetry: 1x (34:01) Most popular number(>9), digit and colour this video: Thirteen (3 mentions) Eight (53 mentions) Orange (61 mentions) Antithesis Battles: High (2) - Low (1) Even (4) - Odd (0) Column (5) - Row (2) 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!

[@jasonhoffarth]

the "ah' at 13:21 is the best

[@WayneDupree1101]

Just wanted to make sure you saw my thanks for the diligent work Inspiring Sand in the statistics I got from your information!

[@felis_timon]

I love how every diagonal was either modular or unimodular. That way to color gave a nice pattern

[@MrWaffles1030]

Even moreso, every positive diagonal is modular and every negative diagonal is unimodular. And in both cases it follows the order green-red-blue left to right and bottom to top.

[@RescueMichigan]

@@MrWaffles1030 I was surprised Simon didn't notice!

[@nattixer]

This day has been rough and knowing I get watch yet another CtC video means everything to me. Thank you ♥

[@sarahnash7174]

Sending you a friendly hug

[@oneeyedman4431]

Another hug coming your way

[@shystudyspy]

I feel the same way sometimes. This channel is really comforting

[@DArtagnonW]

I loved this and am impressed by you, Simon (as usual). It's such a testament to your variant-solving that you so quickly identified the ways in which the two modu-properties were similar. As for the puzzle, I laughed out loud when it became obvious that, not just the 7,8,9-length diagonals, but EVERY diagonal was going to be uni/modular.

[@crumple__]

I noticed around 30 minutes into the video that every single box had identical properties of blue, green and orange. Watching him logic out things I had pattern-recognitioned into felt like cheating. Blew my mind, stunning stuff.

[@ruthphelps1440]

Always a treat when Simon does the intro music. 😊

[@coreydixon9713]

This is the puzzle that finally made me subscribe. I’ve been following the channel since before Covid, but I’m not the type of person that signs-in and subscribes to channels. After finishing this puzzle, I had to sign-in and send my gratitude for this and so many amazing puzzles and apps (love Sven’s SudokuPad 👍). Thank you to CTC and all the constructors!!

[@jonnybolton4659]

This is the only channel I subscribe to!

[@missingdeck9999]

Glad to have inspired you to subscribe:-)

[@HunterJE]

Love that [ending spoiler]... ... ... ... despite the rules only referencing the 7-8-9 length diagonals, the rule actually propagates to every diagonal in the solution, AND every row and column is modular to boot

[@no_name4796]

That's like the easy version of this sudoku

[@minamagdy4126]

I wonder if this can be proven (likely not)

[@HunterJE]

I've done a lot of fiddling in the grid and am almost convinced that the ruleset does not allow for any shading arrangement that does not look like the solution's allowing substitution of colors or rotations (i.e. all unimodular going one diagonal direction, all modular going the other, consistently throughout the grid), though not enough that I'm able to put it into a concise textual proof (would love to see a proof one way or the other if anyone else is able to come up with one)...

[@jordantyner1726]

I have found that given all 7,8,9 diagonals are either modular or unimodular, that all positive 7,8,9 diagonals are all one way and negative diagonals the other.(not necessarily 456) let's start by proving one 9 diagonals is guaranteed to be unimodular. if they weren't unimodular then the center and one of the edges in the center box (let's go with r4c5 for now) would be the same group. with the example square that would make c2 and c8 full of 4 of the centers type breaking sudoku (rows 1 2 8 and 9 in their respective column). this is true no matter how you rotate about the center. therefore the centers type would have to be in the center boxes corners making a unimodular diagonal. we can't have two unimodular 9 diagonals because then we have 5 of the same type in the center box when there are only 3 of the same type. Therefore, one 9 diagonal is unimodular and the other is modular (if the 8 diagonals are modular/unimodular)(this also holds true for different, yet similar reasons if only the 9 and 7 diagonals are modular) similarly once we build the 2 diagonals (let's for arguments sake say the positive 9 is modular and negative is unimodular) the edges of the center box can't be the same type. if they were (let's us r4c5 and r4c6) then that would place 4 of the same type in two rows and columns (row 1 and row 7 in columns 2, 3, 8, and 9 in the example). therefore the edges are different from the corners they are next to. this leads to the 8 diagonals parallel to the unimodular 9 to become unimodular. Also because of the nature of the modular rule the center box's relative modular places are the same as the corners relative modular places. (r1c1 is the same type as r4c4 and r1c7 and r7c7 and r7c1 etc.) now we see that with all the above filled out that the seven diagonals are the only ones left to determine. but with the 8 and 9 diagonals filled out we can look at r3, c3, r7, and c7 to fill out what type the center of the row/column is going to be. from there you can determine the modularity/unimodularity of the 7 diagonals and find that the parallel diagonals are the same modularity/unimodularity. continuing this leads to the fact that without any other info you can't fill out the type for 12 squares. r1c4-6, r4-6c1, r9c4-6 and r4-6c9 unless given an additional clue.

[@minamagdy4126]

@@HunterJE I imagine that a proof would involve, given a valid shading on all cells on the longer diagonals : A. a proof that, along both columns and rows, modular coloring must apply for already shaded cells. B. noting that the only cells still ambiguous are 4 cells each in the even-numbered boxes, one of which is trivially found. If the box 5 logic, as showcased by Simon, can be made sufficiently general, I believe we would be very close to point A. As to how to work with point B, I'll leave that to more active minds. One thing I'll say is that, given the logic that a cell's class must repeat 3 cells apart along the longer diagonals, it is easy to show that all odd-colored boxes must have exactly duplicate coloration. I believe that this is the key to using B given A.

[@intentionalrounding]

There was a two- or three-week spell a while back -wanna say a year ago - where several of the puzzles had this kind of red herring rule set (I.e., in this one, modularity type didn’t really matter and length of diagonal didn’t really matter). Loved those ones back in the day and loved this one just the same. Constructors overspecifying rule sets wrt to the solve is the bees knees of cleverness IMO.

[@Paolo_De_Leva]

What do you mean? In this puzzle, without specifying modularity type, you would not be able to differentiate between *positive slope* diagonals (all of which need to be unimodular due to the *13-cage* in *box 1,* which rules out of itself the *{258}* "module") and *negative slope* diagonals (which can be "modular" due to the *11-cage* in *box 3,* which allows for the *{258}* "module"). For instance, this disambiguation probably would not work with *entropic* diagonals, I guess.

[@nikkisweezea2388]

@@Paolo_De_Leva I think he means the rule set could have easily said 'Every diagonal is either modular or unimodular" rather than stating only the 7, 8, and 9 ones are. This would have made the puzzle much easier, IMO, but clearly it's possible with more constrained, less specific rules. Much like the one the other day where the clue could have been "Every V and X are given" but it was "Not every V or X are given" making the puzzle a bit trickier.

[@laurasmith2173]

Wonderful solve! As the rules were explained, I thought "Oh we get to color!" Then as Simon was understanding the middle diagonals I thought "So the middle cell is shared and has to be a 3." It's good to know my sudoku skills are improving by following the logic presented in these videos. P.S. I have been working through the normal sudoku app and completed several puzzles I never would have attempted a few years ago.

[@Matthew-wz8ng]

Wow, I am amazed that Simon missed two chances of 3 in the corner.

[@Hertog_von_Berkshire]

Became a lot simpler once I renamed, in my head, "modular" as "cyclic" and "unimodular" as "sames".

[@Kamiyurikai]

A very trippy psychedelic colouring puzzle with a cute and interesting solve path. Love how the child in Simon appeared when he figure out the 3rd cell rule trick.😆😆😆😆🎉🎉🎉

[@KittSpiken]

This puzzle and video have been sitting in tabs for the last couple of days and I kept coming back to when I had some free time. I broke in with the central 3 pretty quick, but was too quick reading the rules and applied the every 3 digits of the modular line constraint to the unimodular line as well which placed 3 in the 13 cage on the correct negative diagonal. I pressed on until the puzzle broke, unwound and couldn't see the problem until I came back to the video to watch your start and realized I had added the additional constraint absent from the rules. I paused the video and went back to the puzzle until it broke again and again. I knew I could get it to yield to me, I could taste it; again I returned to the video and you were doing the same things I had done. In the hours and days between working on the puzzle, I had overlooked that I had proven the negative diagonal unimodular with an imagined constraint. I finished my trail and finished the video. Besides clarifying my misconception of the rules and affirming that I was on the right track. I believe this is the longest video length puzzle I solved unassisted. Thank you both for putting this puzzle on my radar, for always encouraging all of us to take a crack at puzzles ourselves and so clearly explaining your thought process in the videos; it is highly unlikely I would have been able to solve this puzzle without following you through some truly diabolical puzzles in the past. Congratulation again on a well earned and well deserved 500k, here's to one million.

[@emilywilliams3237]

So many things to admire about this video, Simon - but first let me congratulate FullDeck and Missing A Few Cards on a truly extraordinary puzzle. I loved watching Simon solve this - thanks so much. Back to you, Simon: one of the things that I totally admire about this channel is the purely logical solves. Assumptions, even if verbalized, are not the basis of the solve, but proofs and articulations of the reasons for things are. This puzzle could have been assumed by the pattern of colors to end up a certain way, or the way the digits fell to result in something in particular - but you proved the solve. That is essential to this channel, I know, and I really appreciate it. Now, to some people, the idea of a "proof" or a logical path fully articulated and explained would seem dull and stuffy, but when you (and Mark) do it, it is delightfully sparkling and fun and even hilarious. "My head's official status is 'done-in'!" "What??!! That's a three by ... GREEN!" and all of the chuckling and throwing your arm across your head and exclamations of enthusiasm - I laughed throughout this video. Clover, if you're reading, THANK YOU for recommending this puzzle to Simon. What a fun evening. (Edited to add - almost forgot to mention the guitar intro music! Always such a pleasure.)

[@missingdeck9999]

Thank you so much for the kind comments!

[@everorizon]

A better way to deduce that one of the main diagonals has to be a unimodular diagonal is to note that, as the corner cells in box 5 are a 6-7-8-9 quadruple, whatever you put in the center cell at least one of the diagonal neighbours has to be equivalent mod 3

[@pixllo]

Yesterday's puzzle I could not solve. Today's on the other hand, 20 minutes. The logic clicked right away, it just felt right. Consider my mind blown by the genius of this setting.

[@missingdeck9999]

Thank you so much! Very glad you enjoyed the solve:-)

[@KevFrost]

If it makes Simon feel better, I can feel him shouting at me far more when I'm solving :)

[@bobduato4643]

My first time beating Simon's time! *not including the countless minutes describing his thought processes lol. I spotted the 3 in the center and every 3rd cell logic pretty quickly and then disambiguated r4c4 to being unimodular using the 258's in r7 and the 147 in c7. The rest just fell into place. What a fun puzzle!

[@sofiagodinho924]

37 minutes in and I'm amazed at how Simon is so focused on the math of it all he doesn't see the pattern of colors repeating

[@th.nd.r]

WHAT AN INCREDIBLE PUZZLE. Hall of famer FOR SURE. Geez Louise. Wow. FD and MAFC, take a bow, and Simon, take a bow. The fact that this has such a beautiful solve path from so little, the unique ruleset, and the revelation at the end that this is modular along ALL positive diagonals and unimodular along ALL negative diagonals is just stellar. Only note: there were two 3s in corners, neither which got a song!

[@stevepinard5826]

@37:42 "It's reduced me to this". Could not stop laughing.

[@abygailwong6022]

Took me 51 minutes and 17 seconds with no hints. Pretty proud of myself, I didn't think I would be able to do it. It always makes my day to solve a puzzle on the channel.

[@seancannon2193]

The final result creates an optical illusion for me that tricks my mind into thinking the orange is green (though it almost seems to fade in an out of doing so for me). It really makes it hit home how difficult being red/green colorblind would be... Still a wonderful puzzle. I appreciate how creating the pattern using only the lines that pass through the middle box propagates that pattern through the remainder of the puzzle.

[@krm1t]

I just had to write a comment since I think that for the first time I actually found the break-in before Simon. And I'm very proud of that :P

[@Coldheart322]

Dispite this being a 46 min video, I actually tried it. So glad I did, some really nice logic which just helps break things apart and you make sudden gains. Then watching Simon see the same logic makes the video all the better. Like moments before 27:45 Simon was trying to prove one thing, then spotted something else, while I was waiting for the penny to drop. Really great puzzle, and really nice video.

[@Coldheart322]

Watching to the end, I had thought there was a secondary rule of "two orthoganally adjacent cells in box 5 can have the same modularity", but actually you can extend to to no row or column can have 2 cells with the same modularity, due to the 3 distance rule putting 4 cells the same in a row or column. Once you spot that, the middle box really falls apart. It is then spotting the killer cages and how they become limited, and the modularity options bouncing back into box 5. I think that forces the entire grid to follow the modular/unimodular pattern.

[@Jodawo]

That was a stunning puzzle. Never saw Simon so distressed or so amazed that he was able to solve a puzzle. I can only think of one other puzzle that gave him more distress and made him more amazed that he was able to solve it and that was "Chaotic Wrogn". If there is another book that is a puzzle that deserves to be in it. What a find. If you noticed it didn't need to go any further than 7 cells of diagonal cells because it ended up all the diagonals lined up in one way or the other.

[@Orenotter]

Diag'nals which have modularity They start to display similarity As we sort them by types They form colorful stripes I must laugh at the utter hilarity!

[@emilywilliams3237]

Love this Limerick!

[@missingdeck9999]

Lovely!!

[@ipudisciple]

Can we agree to call these diagonal lines _Risk_ lines? On a Risk line, every run of three cells is either AAA, BBB, CCC or ABC in some order, where A={1,4,7}, B={2,5,8} and C={3,6,9}. In the game of Risk, there is a similar rule with Infantry, Cavalry and Artillery.

[@sampathkumar-ej7xl]

Simon I hope you do more of such modular rules based puzzles. Some of the puzzles in the modular pack that was released last month were very good and difficult.

[@TracyCarolan]

This is the first of the “unique” puzzles that I’ve bookmarked to do again because it was so much fun! Great solve!

[@missingdeck9999]

We take that as an immense compliment! Thank you!

[@G4M1L]

This was one of the very - very few puzzles, where my time for soiving is not too much more than the length of the video (about 1 hour) - and I'm not a fast solver at all, and I don't think i was ever faster than the video. Key was the discovery that you can easily colour in every 3rd cell on the lines, no matter whether modular or unimodular. That led to a nice colouring of the whole grid, and then the dominoes did the disambiguation.

[@Coyotek4]

The symmetries in the final solution are simply astounding. Offhand, I see ... . . . ... (1) EVERY diagonal is either modular or unimodular [not just those of length 7 or more]. In fact ... (2) EVERY line of positive slope is unimodular, and EVERY line of negative slope; [2a] [this holds true even wrapping around] (3) EVERY combination of 1-4-7, 2-5-8, 3-6-9 appears as a group of three horizontally in ONE box, (4) EVERY combination of 1-4-7, 2-5-8, 3-6-9 appears as a group of three vertically in ONE box [example of 3&4: the numbers {4, 8, 3} appear at r8c123 and r123c6] Simply astounding!

[@kevinshaffer3313]

Building off of your 3rd and 4th observations, you can see that the corresponding box for each group is one box down and one to the right (wrapping around). As you follow the groups across the row, you see that each group is in the same column of the corresponding box. For instance, 3-5-1 in the top row of Box 1 appears in the middle column of Box 5. 6-2-4 in Box 2 appears in middle column of Box 6. It's kind of mind-blowing. This really is one of the most beautiful puzzles puzzles I've ever seen. Kudos to FullDeck and Missing A Few Cards!

[@mega9tales]

43:40 for me and legitimately surprised that I got this on my own. I usually don't get along with modularity very well at all, and it took me a while to get my head around how it didn't really matter whether a line was modular or unimodular, but once I did things flowed pretty smoothly. I was definitely just as floored as Simon, puzzle was absolutely stunning and I found myself laughing as I filled in all the digits at the end too

[@aidarosullivan5269]

Mind boggling symmetry!

[@megalojudge2156]

The first (and probably only) sudoku I've ever managed to solve faster than Simon (and even then only by less than 10 seconds!). Can't pat myself on the back too hard though; modular arithmetic is something I do a lot of, and Simon is slowed down by having to clearly explain every step of his reasoning (which he is marvellous at, by the way).

[@mipsuperk]

When I did the modular lines pack, I used green for 147 (because ctrl-shift-4 makes green, and 4 is in the set), purple for 258 (ctrl-shift-5 = purple), and 369 as orange (ctrl-shift-6 = orange). In this way, the colors cleanly mapped to their own modularity in a weird way. To see a puzzle done with a different color scheme that breaks that logic I did like a million puzzles with is melting my brain.

[@YaBoiRocc]

What an absolutely gorgeous puzzle. The propagating modularity, the striped patterns, the killer sudoku clues disambiguating it at the end. Marvelous

[@missingdeck9999]

Thank you so much!!!

[@amysteele2488]

I'm glad I didn't get to watch this last night when it was first uploaded, as today was a rough day. Watching Simon solve a cracking sudoku like this was just the tonic I needed

[@Teeboned]

I loved this puzzle. Very doable once you understand the modular rules as well. I was yelling about that middle 3 for a bit lol

[@Anne_Mahoney]

I had done the easy set of modular lines by Full Deck, Missing a Few Cards, and friends (on Patreon; this is the generally-available one) and found them an interesting challenge. I haven't taken up the longer second set yet but after this I think I will. This was a fun puzzle.

[@ca-ke9493]

49:17 i wasnt expecting to get it so fast with such a seemingly complicated ruleset

[@pelahnar4]

I watched right up to Simon's eureka moment of of third cell always having to be the same modularity, then said (possibly out loud, I don't remember), "okay, thanks for figuring the basics of this logic, I can take it from here." Then, after I solved it myself, I got to watch the rest of the video gleefully knowing what each next step was and seeing Simon uncover the pattern. I wonder if there's a way to make a sudoku grid with these constraints that _don't_ result in a striped pattern like that. I could definitely see these constraints making only two patterns (one where the positive diagonals are modular and one where the negative diagonals are modular) and, what, 27? different puzzles for each? Based on how the digits resolve. So, 54 possibilities total. But I'm not sure.

[@olivier2553]

I think that you can only have that striped patterns, I saw that instinctively should be it from the very beginning, even if I would have been unable to prove it.

[@mawillix2018]

I tried to calculate it, and it seems there are a total of 72 permutations. (This is before I found out the solution to the puzzle, so it surprised me that there weren't more permutations.) After looking at how it solved here, it seems that 72 permuations is correct.

[@mawillix2018]

The above might be incorrect, I'll calculate it again, I feel I missed something.

[@mawillix2018]

Ok, I found the *first* problem, I still needed to choose the order the other two sets of 3. That's 72*36=2592 then. Actually, the entirety of the permutations is: Choose a main diagonal passing through the central cell, it is now unimodal. (2 options.) Choose the color of the central cell. (3 options.) Choose a different color for an orthogonaly adjacent cell. (2 options.) Order the three digits in the central box that are on the unimodal main diagonal. (3*2 options.) Order the digits that are orthogonally adjacent to the central cell. (3*2*3*2 options.) That's 2592 permutations. Now I need to check if there's something else that can be changed... and there was. After solving this far, each color solves similarly to a 3 by 3 sudoku with their central cell filled in. (Same amount of permutations, which is 4.) 4*2592=10368 different permutations in total. (Meaning 72 was way off)

[@mawillix2018]

In the end I forgot that *each* color solves like a 3 by 3 sudoku with their central cell filled in. 10368*16=165888 permutations.

[@Ulija100]

This is very special, thoroughly enjoyed it. Thank you!! Surprised to be able to solve it. Wasn't pretty the first time, but then could loop back and redo a few times to really understand how it works. Then joy of watching Simon uncover all the keys!

[@fluffycritter]

That puzzle was absolutely fantastic. I think I can see the logic to how it was constructed in the first place, too, especially with the full coloring at the end. It almost feels like the discussion of 'modular' vs. 'unimodular' was a red herring to make it seem more complicated than it is.

[@phileo_ss]

I genuinely thought the logic was beautiful. Brilliant puzzle and solve.

[@andersosterholm2538]

Wow! That's one of the most amazing puzzles I've seen! Thanks to the setters and Simon for making this video possible! It was a joy to watch.

[@missingdeck9999]

Thank you!!!

[@TeddyBear1287]

Just finished this myself w/out watching any of the video... LOVED THIS PUZZLE!

[@bristolrovers27]

Great puzzle, well done to the creators and Simon for getting the break-in, which I completely missed

[@miran248]

Speechless! What a beauty!

[@SirBradiator]

Stunning Puzzle 22:47 for me, once I understood the break-in it flowed very easily, loved it

[@ericpraline1302]

Gratifying to get through this one as it's the sort of puzzle which really stretches my creaky old brain. Well done brain, you've earned a lie-down in a darkened room.

[@sebastienlecoq3956]

0:40 : Clover has said this is a puzzle Simon should attempt 15:35 : the way to do this is probably to color Clover knew !

[@KevFrost]

3 hours and 4 resets later ... I have neopolitan icecream colouring pattern, wonderful puzzle.

[@rgoyal107]

How interesting that it wasn't just the 7, 8 and 9 length diagonals that had the property of being either modular or unimodular, but every single diagonal of any length has the same property.

[@femto113]

This one solved quite fast for me (~10 minutes, easily the fastest I've ever done a CtC puzzle) because I quickly stumbled onto the fact that there are only two ways to color the grid: the unimodular stripes either go top left to bottom right, or bottom left to top right, and the two dominoes on top only work one of those ways.

[@thefallenarm589]

17:00 I explained this to myself as : the central digit is connected to all four corners ; one of the four corners at least is of the same modularity so at least one of the main diagonals is unimodular (which needs to 2 corners of he same modularity, etc). Then, all four corners are not of the same modularity, so one of the main diagonals is modular.

[@johninnaperville]

After doing the puzzle hunt based on the modular sets, it was easy to see where this was going. They make great colorful sudokus as it is almost impossible to solve them without coloring. Simon chose the same colors I used but for different sets of digits and I wanted to scream, NO blue is 3,6,9, but of course he didn’t listen. It is fun, when you have the grid totally colored, to,just pop in the digits at the end.

[@Baritocity]

It's marvelous how few cages it took direct the modular part of the puzzle the way it does.

[@zdenekpavlatka609]

Amazing puzzle, I now feel bad for not trying it before watching the video

[@aere481]

Great puzzle. I managed to get the breakin quickly. This channel has taught me so much!

[@MrRietmann]

Great intro !!

[@flackbyte]

There's something quintessentially English about the expression "sorry for the fist-pump".

[@willemm9356]

In retrospect, the colour pattern of the central box will always be completely reflected in all the odd boxes, from which almost all the logic can be deduced. Most importantly, no two of the same colour in the same row or column in the box.

[@stevezagieboylo9172]

That had to be the hardest way ever to prove that one of the main diagonals is modular and one isn't. How about this: Of {6789} which we know are on the diagonals, one of them has the same modularity as whatever we put in the middle.

[@robert-skibelo]

Bravo. I have the feeling that's not the sort of thing Simon notices. The need to talk all the time reduces the opportunities for penetrating analytical insights.

[@stevezagieboylo9172]

@@robert-skibelo To be fair, I posted because it is one of the very few times I spotted something that Simon didn't. The most common situation is that I've played until I'm completely stuck (having taken much longer than he does) and I have to play the video to get a hint from his insights. Then I pause and continue. It is only about one puzzle in three that I complete without this sort of hint.

[@wanderlustwarrior]

55:10 Wow, that was a really entertaining challenge!

[@missingdeck9999]

So glad that you liked it!

[@hummakavula3750]

I was just a few steps ahead of Simon and had to laugh when he found the 3 and said it wasn't worth a song. What an incredible break in.

[@polill00]

I'm excited to try the new pack :D

[@skid68]

That was a impressive puzzle and great solve Simon :)

[@simonedgar7566]

"That's a 3 by green!" - new one for the list :P

[@missingdeck9999]

Really enjoyed that new Simonism:-)

[@Raven-Creations]

Very nice, and I can see why Clover in particular would recommend it. There's a certain Cloverishness to it - not too hard, just really nice, clean logic. 21:04 for me. I think this is the first time I actually felt smarter than you, rather than just beating your time by using a more disciplined approach and being more observant. I've not done a puzzle like this before, so familiarity wasn't a factor. The key thing to note is that regardless of the type of diagonal, the third digit along always comes from the same group, and by extension, the edge cells of box 5 have two cells in the nearest row and/or column which match. In particular, the corner cells of box 5 project two cells onto both the nearest edge row and column. Your fear of pencil-marking slowed your start. If you'd pencil-marked 6789 in the corners of box 5 sooner, you'd have spotted that 3 had to be in the centre with a modular line going one way through it (with 78) and a unimodular line going the other (with 69). This is because you've got two 369s and one each of the other groups, If you try to put the 6 and 9 on different diagonals, the central cell is common to both, so the other two would have to be in the same group, but 7 & 8 are in different groups. This means you can put 369 in the centre cell of each corner box, putting 258 into the top of the 11 cage, and 47 into the left cell of the 13. In turn, these fix the types of R4C5 and R5C4, and then these bounce back to the edges. You now have two 258s in R1, so neither of the top corners in box 5 can be 258s, or you'd have four in the row. Similarly, you have two 147s in C1, so neither of the left corners of box 5 can be 147s. Crucially, this leaves you with a 69 in R4C4, so you can tell that the negative major diagonal is unimodular, and the positive is modular. There was no need for any of your bifurcating, saying "if this is orange, then this is orange, and this...etc." Instead, you could just note that two of a colour in a row/column prevents that colour appearing again in the row/column three rows/columns away in any cell touched by any of the key diagonals. E.g. just like you eliminated blue from R5C6 because you had two blues in R2, you could have applied the same logic for all cells. If you had set aside your scatterbrain approach, and stuck to doing it methodically, you could have raced through it.

[@osirusbrisbane2305]

Tremendously fun puzzle! Wasn't sure I'd be able to spot the break-in, but once I did, everything came together beautifully.

[@johnargeles7019]

amazing puzzle! love watching your videos before going to bed

[@DanFletcherVideo]

That’s THREES! In the corner. I can’t believe we didn’t get the song when there was two opportunities for it! :P

[@jessicalenoir9979]

The guitar at the beginning made my entire day.

[@randomstrategy7679]

Finished it in 21:07 for me. What really helped was just considering this to be a sudoku with only 3 digits to be placed, each of which appears 3 times in every row/column/box. Then proving that the five "odd" boxes were a) identical and b) each was a mini-sudoku with every digit appearing once in every row/column. After that everything was easy.

[@Blueforest20]

took me 28:31 it was all my practice doing the patreon modular line pack ingrained in me forever

[@JosueSantiagoG]

I was just noticing last night that it's been a while since I'd seen Simon play the intro. Thank you!

[@Tringard]

Very satisfying solve, especially as the patterns started to emerge and how disambiguation works. edit: having watched the video, I didn't start putting digits in beyond the center box until nearly all of my grid was colored in.

[@leickrobinson5186]

Took me 57 minutes, but I’m perfectly happy with that for a puzzle like this one! :-D

[@Coyotek4]

52:56 ... I made mistakes on multiple pass-throughs which crushed my time. Worth it to see the finished product. Incredible puzzle!

[@mstmar]

i found a trick that blew open the puzzle. using simon's 3 away are the same color trick, if you take the x'th cell in boxes 1, 3, 5, 7 and 9, they all have to be the same color. e.g. the 4th cell in those boxes are r2c1, r2c7, r5c4, r8c1 and r8c7 are all the same color. which is really important since in columns and rows 1, 2 and 3 each color comes in pairs. this means that each color in each row and each col of box 1 (and so each in the other odd boxes), you get 1 of each color. so in the central box, 6 and 9 have to be in opposite corners (or you'd have 2 greens in a row/col) making the center cell a 3. you can then follow the vid from 20:00 to use the dominoes in box 1 and 3 to color r4c5 and r5c4. then by the trick, you can color the rest of box 5 (and so all the odd boxes). From there the puzzle is simple to solve

[@sofiagodinho924]

a truly beautiful solve

[@SilverEchoes]

This was marvelous! I can't claim to have even an inkling of the underlying mathematics at play here, but I could see the workings of those laws nonetheless. This is a stunning embodiment of math, a visual and tangible display of not just logic but the magic of numbers. The taste it gave me leaves me wanting more; I'd be fascinated to learn why and how this works, why all these patterns emerge with modularity.

[@missingdeck9999]

We are really glad you liked it! At some point soon, we'll put out an explanation of why this works.

[@SilverEchoes]

@@missingdeck9999 I'm looking forward to it!

[@SylwiaPoland79]

New video. Guitar. Sudoku. Perfection 😁

[@whatsleep17]

Amazing puzzle!!!!!

[@noahvale2627]

Wow. Just wow.

[@ArunIyerS]

What a clever puzzle and a very brilliant ruleset 🙂

[@ProfessorPanyck]

I'm sure there is some 3's rule going on, but it's fascinating to see the colored grid be identical in every square.

[@brucejones533]

Amazing!

[@ChrisVenus]

One observation on the middle box and the pattern is that two colours in the same row or column would be impossible due to the thing you found about three away diagonally being the same colour. If two cells are in the same row then of course the ones three away diagonally will also be in the same row which means that if two were the same colour you'd have four of them in that row. Same of course for columns.

[@davidturk5846]

Such a cool design.

[@asafsh4]

i usually watch CtC to relax after a long day at work: solving on my own, watching a bit if i get stuck, and then watching the rest so i can see if we did it differently but this time... watching Simon finish coloring it all and then NOT solve them by color and insted jumps around made my OCD brain explode (i also did it with the 369s then the 258s and then the 147s by reverse order but i would be ok with different orders) not to speak of the fact thay he had all the domino pairs after the greens but he claim that the 7 was "MAGICAL"?!?! still had fun with the puzzle and the video though :)

[@ceevio_art]

Same here. Its like Simon "discovers" simple sudoku logic every puzzle he does, as if he's never done on e before. Very weird.

[@douglasrogers3918]

That Simon was amazing! Picking up the logic of every third cell being the same modularity when you were not familiar with this concept is just brilliant. Take a bow.