A Difference Of Squares: A Sudoku Masterpiece

[@mathguy_1243]

Wow!!!! Thank you so so so much for solving my puzzle, I was nearly in tears with happiness of being featured! And to not only be featured but to receive such high praise has made my birthday that much more special, thank you again so much. It was such a surreal feeling watching Simon solve my puzzle and it was such an excellent solve at that, I will never forget the expressions you made at breakthroughs!

[@efad3215]

Happy birthday!!

[@patrickgass787]

High praise and well-deserved! Looking forward to your next feature already

[@Alex_Meadows]

Best wishes for a very happy birthday, and what a fantastic debut puzzle!

[@frankwilliamhansen9550]

Happy birthday, what a puzzle! 😍

[@darreljones8645]

Pity your new age, 22, cannot be expressed as a difference of squares. 21 can, as (5*5) - (2*2).

[@brianpso]

I came from numberphile to see your original solve and I'm so impressed with how quickly you managed to figure it out! I wouldn't have been able to find this in a lifetime!

[@Pulsar77]

The best thing about this channel is that the more puzzles you watch and try yourself, the better you become. I've been doing this for two years now, and at first this puzzle would've been impossible for me, but now I'm usually able to solve even the hardest puzzles on the channel, although I'm typically 2 to 3 times slower than Simon 🙂

[@FirstLast-gw5mg]

I'm familiar enough with the Phistomefel Ring that I spotted what I call an "offset Phistomefel Ring". Basically if you take a regular Sudoku (no extra rules), you can take columns 1-3 and move them to the right side without violating any of the constraints; every row, column, and box will still contain the digits 1-9. In a similar way, you can also move rows 1-3 to the bottom. If you do both of these transformations to the Phistomefel Ring, the result is the same two sets of highlighted cells that Simon had at 18:05 if you'd add the arrows back in. But then removing the cells along the arrows - their sum is in one set and the digits adding to their sum is in the other - is the key to finding the mathematical equality at the heart of this puzzle. Alternately, the Phistomefel Ring's proof can be repeated, but using rows 5-8 and columns 1-4 & 9, then remove box 1 from the larger set to make it equivalent to the smaller set, resulting in the offset square pattern.

[@strangest_cacti]

This is undeniably my favorite maths-focused break-in featured on this channel. Watching Simon figure out the meaning behind the title and the value of X-Y has been incredible 🙂

[@mathguy_1243]

Thank you! I too loved his expressions at breakthroughs

[@spauldine]

Simon, you warmed the heart of this retired algebra teacher. I love to see someone getting so much pleasure by factoring a difference of squares. And yes, I would love to talk to you at a party.

[@glum_hippo]

You're doing it now! This is the only party that Simon is invited to at the moment.

[@LuckyLuke79a]

Brilliant puzzle and brilliant solve by Simon who never stops impressing me. 😎 But Simon, please. You put so much effort into explaining every step of the solution to the viewers, but then you rearrange quadratic equations in your head just like that. Don't be shy to pull a notepad on the screen to visualize your thought process. It doesn't take away from the beauty of the puzzle and the solve, but I'm sure, people would appreciate it. Thank you ❤️

[@LevelTUV2]

Not sure how much editing they do, but they could also add something in post.

[@Arcessitor]

Yeah I understood what he was trying to do but was completely lost in his visualization of the algebra. I should've paid more attention in school.

[@Paolo_De_Leva]

@@LevelTUV2They know how to post-edit. Simon did it several times in the (not recent) past.

[@Paolo_De_Leva]

I totally agree. I posted a similar comment below.

[@amoswittenbergsmusings]

Perhaps the Unwritten Ethics Rules of Discovery, Elucidation and Intelligibility frown upon reliance on Patriarchical Algebraic Notations. The High Wizengamot of the Kounsel of Kriptik Krakking has taken a "Do Not Fix It If Is Ain't Broken" attitude thus far.

[@Paul-cn3ij]

Ok. I try to stay away from overusing superlatives to describe anything but this is one of the best puzzles I've ever solved. Genius.

[@mathguy_1243]

Wow!! Thank you so much!

[@Paul-cn3ij]

@@mathguy_1243 My pleasure.

[@Tsunami14]

I was NOT expecting the solve path to lead into an algebraic equation. The endless creativity shown by these puzzle setters is absolutely astounding!

[@mathguy_1243]

Thank you! I love the idea of making unique puzzles and I was pretty sure nothing like this had been done before :)

[@Tsunami14]

I just say your puzzle featured by Mark on Numberphile. Congrats!!

[@inspiringsand123]

Rules: 04:11 Let's Get Cracking: 05:54 Simon's time: 46m43s Puzzle Solved: 52:37 What about this video's Top Tier Simarkisms?! Bobbins: 3x (39:16, 42:46, 50:40) Maverick: 2x (41:32, 41:32) Three In the Corner: 1x (49:04) And how about this video's Simarkisms?! Ah: 15x (22:47, 22:47, 22:58, 23:12, 23:12, 23:37, 23:37, 24:13, 26:52, 31:23, 36:30, 42:22, 48:54, 50:40, 51:51) By Sudoku: 9x (24:01, 31:38, 34:20, 34:54, 39:54, 44:42, 45:28, 49:45, 51:18) Sorry: 7x (04:11, 10:19, 33:32, 42:46, 43:57, 46:23, 52:02) Obviously: 7x (09:56, 13:06, 13:41, 14:01, 15:37, 36:30, 38:08) Hang On: 6x (11:17, 16:25, 24:59, 33:07, 33:24, 43:39) Brilliant: 4x (34:15, 41:45, 52:37, 52:41) Good Grief: 3x (16:22, 19:48) In Fact: 3x (29:13, 46:45, 49:34) Out of Nowhere: 2x (35:28, 48:02) Nonsense: 2x (15:34, 15:37) Beautiful: 2x (37:38, 53:42) Incredible: 2x (02:23, 02:23) Our old Friend Sudoku: 2x (35:49, 48:17) What Does This Mean?: 2x (35:05, 44:57) That's Huge: 2x (34:09, 49:38) Pencil Mark/mark: 2x (30:06, 30:19) What on Earth: 1x (04:41) Goodness: 1x (40:03) What a Puzzle: 1x (52:32) Bother: 1x (43:03) In the Spotlight: 1x (49:07) I Have no Clue: 1x (12:43) Stuck: 1x (39:41) Lovely: 1x (53:34) Ridiculous: 1x (19:42) Gorgeous: 1x (18:15) Hypothecate: 1x (43:31) Shouting: 1x (03:18) Masterpiece: 1x (00:49) I've Got It!: 1x (33:29) Disappointing: 1x (32:39) Whoopsie: 1x (36:42) We Can Do Better Than That: 1x (19:15) Progress: 1x (32:56) Plonk: 1x (31:45) Wow: 1x (27:30) Have a Think: 1x (16:37) Most popular number(>9), digit and colour this video: Sixty Four (8 mentions) Two (98 mentions) Blue (32 mentions) Antithesis Battles: High (3) - Low (2) Odd (3) - Even (3) Row (13) - Column (11) 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!

[@WarWithVarun]

Thank god this is a bot lmao

[@lavanyasalagrama7339]

The breakthrough to x^2 - y^2 = x + y is absolutely brilliant! Excellent work, Simon!

[@258thHiGuy]

As a math major, this break-in was absolutely delightful

[@TheSonicPerson]

I agree. I was also a math major in college

[@mathguy_1243]

That makes three of us at least (tho I’m guessing there are probably a few more, need more evidence to back up my claim 😂)! I’m glad you liked the puzzle :)

[@JustWillB]

Me too. Well Done @mathguy_1243. Lovely puzzle.

[@markheclim2741]

As someone with a math minor, this reminded me too much of the 'proof' that all numbers are equal. I was extremely concerned for a moment when Simon divided by (x + y), but then remembered the constraints on x and y that prevent the issue.

[@johnpauladamovsky86]

This puzzle is a geometric work of art, and one Hell of a debut... I wonder how long it took Nathan to wrestle it out of God's mind?

[@mathguy_1243]

Thank you so much!! The math is the basis for a research project that I started in high school around 2016 and is still going in today, the puzzle started to be set in June of 2021 and had gone through many different iterations before landing on this version in October of 2022. It then took a year for me to feel confident enough to send it in to CTC lol

[@johnpauladamovsky86]

​@@mathguy_1243 That's a great story... Some of the artwork puzzles on this channel have the feel of maybe a lightning strike of inspiration, followed by a day or two of working out the loose ends, but NOT your puzzle, it was far too sophisticated and intricate for that type of genesis. It was properly signaling toward the break-in, but even then, the sets were not so easy to spot, and after that when the titled equation unfolds, it just seems like the kind of perfect-balance which took years of perspiration to fine-tune. When I graduated high-school in the year 2000, I was a mathematics prodigy. Then I became an aerospace science engineer, then an artificial intelligence deterministic algorithm designer, followed by a competitive feats of strength athlete, and artist. I am also known for mounting the most spectacular Pro-Se legal defense of the 21st Century, and Jordan Peterson once compared me to a young Arnold Schwarzenegger. Over time, I also became a connoisseur of words that go together well. I'm always trying to find clever and funny ways of recapitulating the eternal verities. I would like to know more about you, because I think you have great potential. Reply if you enjoy listening to masterclass rhetoric and clever witticisms... When Simon read out my name for donating to the first greatest hits book, he said this, "Thanks to Saint Johnny von PaulyMath... One of the Great Names". That was an amazing feeling when Simon complimented me on the nickname I gave myself, so I can only imagine how sublime it must feel to have Simon sing your praises the way he did yesterday... Either way, thanks for replying to my comment. Go out there and do great things...!

[@sskar9390]

Numberphile people here

[@michaellautermilch9185]

Impressively fast solve!! I wasn't able to "work ahead" at all without pausing in this one. Usually I can at least scan for a digit or two before you find it.

[@sillvvasensei]

I love math, so this is probably my favorite puzzle on the channel to date.

[@mathguy_1243]

Thanks you! Definitely my favorite video too 😂

[@Sujisan4]

This was the best puzzle ever solved on the channel.

[@mathguy_1243]

Wow!! Thank you!

[@Raven-Creations]

Around the 30 minute mark, you're making heavy weather of this. The arrow in box 8 cannot be 9, or you'd have 942 in the 15 cage, and 35 in the Y cage, but Y=7. The arrow can only have one of 2345, and the other must be from 678, so it must be 26. This makes the Y cage 34, R9C6=5, R9C5=7, R9C7=8, and R8C4=6. The 15 cage can only be 258. You didn't spot that there was roping in columns 4-6, which would have saved some time. Very clever puzzle. The break-in was excellent. The way the arrow in box 9 eliminated all of the non-caged cells from the two sets was brilliant. After the break-in, the flow was really smooth too. It's not often that such a devious break-in is followed by such a smooth path. Most either fall apart completely, or get a bit messy.

[@Merione]

Only two months ago?! I just came back to rewatch this "old" gem after the feature on Numberphile and it's just two months ago?! Why does this feel like a classic already?

[@joadbreslin5819]

"Good grief, I've done it." Immediately followed by a cut to advertising. What a cliffhanger!

[@mathguy_1243]

IKR!!! There were several moments like that when I first watched it

[@Felazza]

who else is re watching this because of the Numberphile video?

[@thefallenarm589]

21:30 Simon is refering to the 3rd remarkable identity (I'm french so not very confident on the translation) a² - b² = (a - b)(a + b) [SPOILER ALERT below] so in the puzzle we have X² - Y² = X + Y so we can directly deduct that (X - Y) = 1

[@mjschryver]

"we can directly deduct that (X - Y) = 1": How?

[@heinosackmann5599]

@@mjschryver As Simon explained in words, the left side of the equation is X² + Y², and that's X² - Y² = X² + XY - XY - Y² = (X + Y)(X - Y), If you then divide by the right side of the equation X + Y, then there is X - Y = 1. And you are allowed to divide by X + Y, because they are both positive.

[@FirstLast-gw5mg]

This is not entirely accurate; X = -Y is the other solution to the equation. However this equality would imply that either the X or Y cage would have to contain a negative sum, which is impossible since the Sudoku cannot contain negative digits. X^2 - Y^2 = X + Y (X + Y)(X - Y) = X + Y *If* X + Y is not equal to zero, then you may solve as follows: X - Y = (X + Y) / (X + Y) X - Y = 1 However, if X + Y *is* equal to zero, then you may not factor it out, and you must instead solve as follows: 0*(X - Y) = X + Y 0 = X + Y X = -Y Both solutions are valid, and both must be considered unless another constraint invalidates one of them. In this case, the Sudoku's constraint eliminated the X = -Y solution.

[@literalstardust445]

an insanely brilliant puzzle, and a 3 in the corner? absolutely amazing. happy birthday MathGuy_12!

[@mathguy_1243]

Thank you so much!

[@poppyholly1759]

Not seen the solve yet. Surprised (and pleased) you're reviewing Hi Ren. He articulates mental health and social issues very well, and movingly. Looking forward to your reaction especially as your channel has an incidental beneficial effect on mental health.

[@Alex_Meadows]

I'd never heard of Ren before today, but your comment prompted me to google him and now I've watched Hi Ren. What an extraordinarily talented young man, and I do hope that Simon releases his reaction video.

[@poppyholly1759]

@@Alex_Meadows He is amazing isn't he.

[@poppyholly1759]

Just watched the solve and enjoyed it. Usually watch a CTC video each day. Don't know how you maintain the quality and quantity of output. Thank you.

[@MatthewLenton]

I wasn't familiar with Ren although his "version" of Bittersweet Symphony came up in my recommended feed a few days ago but I forgot about it I've just watched a few of his music videos

[@emmbeesea]

I'm a sucker for wordplay and logic puzzles. Hat's off to you, MathGuy_12!

[@mathguy_1243]

Thank you! It’s definitely the best puzzle I have ever set

[@EpicNicks]

Surprise e

[@hypotheticalhippo2327]

Hello to all the people coming to (or back to!) this video from Simon's Numberphile video!

[@morimcore]

this is one of my favorite break-ins i’ve seen on the channel. that moment of realization was just so cool! i’ve always been intimated by math-based puzzles because i’m terrible at math, but this makes me want to try more of them :)

[@mathguy_1243]

Wow thank you so much! Watching Simon break in on set puzzles in the past was part of my inspiration for this puzzle and I’m so glad it was able to inspire you!

[@bradk1406]

Watched the numberphile video and had to come back and watch this!!

[@scottrs]

Came in from numberfile. New fan

[@Seth_M-T]

Absolutely beautiful!

[@TheLukeskywalker2]

I know I solved this when you first solved it, but I'm back from Numberphile video

[@tommy_salami108]

Came from the numberphile video. NIce!

[@sealed2mybff]

Woohoo! I finally have time for a longer puzzle! I've missed you, Simon!

[@sealed2mybff]

Oh my word! It's been so long since I've had time to sit down and do a Simon Sudoku, I forgot how incredibly difficult they are and how freaking brilliant he is!

[@xtra6836]

i love both of your renditions of 3 in the corner, sometimes i find myself trying to guess which one will be sung

[@amoswittenbergsmusings]

Notepad, or no notepad, that is the question: Whether 'tis purer in the mind to figure The squares and arrows in such delicate balance, Or to lean on written stuff, as transient mind extension And on some sheet compute them. To write-to solve, No more; and by a solve I mean we get The answer and the right equation That cracks this mindfruit of a young constructor So expertly displayed. To write, to solve, To solve but not in purist fashion? Nay, do not Ever leave the righteous path of cracking Without the aide of other than the ruminations Of mortal flesh and intercranial cunning. ‘Tis thus That Nathan’s brilliance brighter shines And Simon’s sheer delight is truest manifest.

[@Orenotter]

That puzzle was totally great! Even though I could not quite equate. Imagine my Heaven When finding Y seven And somehow or other, X eight. (Leaves a virtual cupcake topped with two strips of nori)

[@mathguy_1243]

Beautiful!

[@maxcalderon7815]

Numberphile feature numberphile feature :D this really was such an awesome construction

[@jaega4247]

Glad I didn't attempt this puzzle. My math is awful, and it's such a delight watching Simon in his right element. Beautiful puzzle as well! It's just not for me to handle.

[@mathguy_1243]

I too loved watching Simon solve! Thank you :)

[@jamesdonnelly9217]

Thanks so much Simon for bringing Ren to my attention. He's such a musical genius yet I was unaware of him.

[@HunterJE]

Interesting to note, "X and Y are positive integers such that X>Y" is doubly superfluous/self-proving in light of the clues in the grid: -"X and Y are positive integers"-"X" and "Y" cage sums would be impossible if they weren't -"Such that X>Y"-the two "X-Y" cage sums would be impossible if it weren't

[@HunterJE]

("Digits may repeat in cages if allowed by other rules" is also redundant given the 11-cell cage but imo doesn't really function as a rules statement so much as a reminder that the previously stated rule did _not_ include that part of normal killer cage rules)

[@samuelhunt9265]

Sneaky that the only numbered cage in the grid is also x + y haha

[@AFT3RDAY5]

It is so impressive that Simon does all that math in his head 👏🏻👏🏻👏🏻👏🏻👏🏻 I have to scribble it down and am still slower than him 😅

[@HP4L16]

Managed to solve this without knowing the sudoku theorem (so no set theory whatsoever) or using bifurcation, just pure math, so it took me hours. Even though I knew it had something to do with x^2 - y^2 = (x + y)(x - y) I couldn't see what it did without the theorem so all I could do was eliminate possibilities until I knew x = 8 and y was 7, and eliminating the 6 was not easy, had to use a combination of math and sudoku to figure that out.

[@SirCumference31]

(x+y)(x-y) = (x+y) Another solution to this is x=-y which is obviously impossible here since x and y are positive but it's worth pointing out since it's a very common algebra mistake to divide both sides by an expression without considering whether it can be equal to 0. I wonder if Simon thought about this when arriving at x-y=1

[@MegaTrain]

That possibility is eliminated by the two cages totaling X-Y, which can’t be zero, so x cannot be y.

[@SirCumference31]

​@@MegaTrain x=-y is the other solution, not x=y. It's easy to eliminate x=-y as explained above.

[@RemkoNHN]

Oh wow.. Never heard you talk about Ren before. Just came from his stream :) Amazing artist! Can't wait for the reaction video! Will that be on another channel?

[@davidmeijer1645]

Simon, you are one clever fellow, for 1) noticing the double meaning of the title, 2) doing that algebra in your head, (I’ve been teaching math for 30 years and knew this, but for you to conjure it up like nothing, having not seen it in a while, kudos! 3) all that cancelling of set of digits, brilliant! But explain to me why the 3 blue square cancels the corner square, (you said; say it was an 8). Why?!?

[@mathguy_1243]

Hi! Just clarifying why the bottom right cells cancel: first Simon tells us that before any canceling is done that the cell appears 2x in green and 1x in blue, we can cancel 1x from blue and 1x from green leaving it 1x in green. Then, because of the arrow clue, we know that the bottom right cell has the same sum as those three blue cells and since at this point we don’t care about conserving exact digits between the two colors rather but exact sums between the two colors if we subtract the three sum of the three blue cells from the blue set and the sum of the bottom right cell from the green set the two sets still have the same sum since we subtracted the same amount. Hope that helps!

[@wrightn9]

Why exactly is it impossible for Y=8 and X=9?

[@Tamanna_2812]

Because the maximum the y^2 cage can go is 54, and thus y=8 is impossible as 8^2=64

[@emilywilliams3237]

This was delightful, Simon. I texted my son as it is the kind of puzzle right up his alley, I think. I am always appreciative of your proving the various set theory ideas in these puzzles, as I think I learn them better each time.

[@emdiar6588]

Simon, PLEASE upload your Hi Ren reaction. It is an unmitigated masterpiece, imo, and I'd love to see your first and genuine reaction to it. None of the swear words are gratuitous and are all integral to the power and poetry of the piece. Maybe a strong warning at the start would suffice, because dubbing a ''bobbins'' over them all might be considered the act of a philistine. :) :)

[@tangrybird04]

1:15:11 for me 😅 I literally wrote out the equations on a piece of paper.....I couldn't properly visualize anything,,,,,,great and beautiful puzzle

[@mathguy_1243]

Thank you so much for solving and your kind comments!

[@tangrybird04]

@@mathguy_1243 keep doing what you’re doing my guy 🙌

[@psythagoras113]

@mathguy_1234 as a teacher of Mathematics for over 20 years and a lover of Cryptic Sudoku, this has to be the greatest break in that I have ever seen. Just incredible!!!

[@mathguy_1243]

Wow thank you so much!! The math itself is super interesting and there’s a lot more complexity to it when you start taking things to higher powers, it ended up being my sr research in college

[@mikew6644]

Some of us don’t like “proving its position”? Who?! How?! Why?!

[@bobfish7699]

Video Featured on NumberPhile.

[@BryanLu0]

I'm sure they know, it is Simon in that video

[@TheHajsan]

This is truly impressive, from Simon and Nathan alike. A brilliant video/solve and a brilliant puzzle. Well done!!

[@amoswittenbergsmusings]

Apologia Pro Vita Sua This the confession of a prodigal son. My fracturing mind has taken an extended sabbatical in the land of doldrums and I have failed to take my digital medication at regular intervals. The universe has however not stopped singing and Simon still has the proper aerial and amplifier. He popped up in my feed as Brady's guest, twice. I just had to see this solve. Brethern and sistern, please forgive my faithlessness. I cannot reliably promise regular comments. My mood swings tend to be extreme these days.

[@Afterthoughtbtw]

Embarrassingly, I had to google search what the quadratic formula was in order to solve this puzzle. Had completely forgotten... and I was good at maths in school, too! Still, I managed in the end.

[@pooppoop3595]

I dont think that’s embarrassing! Just means you havent used it in your day to day as of late

[@room5245]

Watched a few of yours years ago, super inspirational seeing geniuses at work! Now back because of the Numberphile vid, funny to see your first instinct was totally right!

[@rafael_l0321]

out of this world. wow

[@huwkin]

Incredible solution. Pleasure to watch you every step of the way

[@saiyanprince989]

Another thing that could have helped a bit is to realize that columns 4, 5, and 6 has roping.

[@Agent29416]

Here from numberfile kinda funny him instantly saying bo that isn't helpful when it was how he demonstrated it on numberfile. But i assume past you will get there

[@chesterfield9682]

You're right, some of us don't like the "proving its position". It is "losing its religion", and I will die on this hill.

[@wanderlustwarrior]

1:39:24 for me, that was quite challenging.

[@mathguy_1243]

Wow! Way to stick through it and thanks for solving! :)

[@JohnRandomness105]

I needed the set-theory part to break into the puzzle. I didn't know it, and I didn't know anything else to do. I think that I managed to do the rest on my own, with considerable difficulty. 30:50 At one point, I did rule the 2367 into the particular cells of the cornermarks: R9C3 had to be 67. But that was a little later. I don't think I used the gigantic cage any further in the puzzle. Sudoku and the other constraints forced the digits to sum to 64.

[@mathguy_1243]

Thanks for solving!

[@DasBadedyr]

@simon, the puzzle named "the fourth killer" also had a genius name 😉

[@sakkikoyumikishi]

It was very amusing to watch Simon argue why the lower lefthand arrow can't sum to 9 twice, then move on to try and figure out whether *that same arrow* sums to 8 or 9, trying to find a reason why it would *have to 9*. Especially when he then pointed out - for the third time - that there can't be a 9 in the 15-cage because that would remove all options from the y-cage. But still didn't realise that therefore no 9 could be in the arrow

[@alexperlinger]

Once you got your equation at 20 mins, I recognized the pattern immediately. When green=blue, green=y^2+y+x and blue = x^2, you can use n^2+n+(n+1)=(n+1)^2 with y=n and x=n+1 in the case of this puzzle.

[@johnfsenpai]

I turned 22 a week ago, it's incredible to see there are people my age who are that smart

[@DanielHShead]

I really enjoyed this puzzle. The name is going on my favorites list next to Syzygy and Subtle Advertisement.

[@mikesommers1068]

Surely Simon didn't need the complex equation? The puzzle itself tells us- in 2 different cells-- that x-y must equal maximum 9. Also that x and y can be the sum of two digits so must be a minimum of 4 and 3 respectively. The only squares with a difference of 9 or less that qualify therefore are 3/4 or 4/5, both with a difference of1.

[@mikesommers1068]

ignore me, I'm talking rubbish Doh!

[@kgbgb3663]

This is the first variant sudoku that has delighted me so much that I managed to persuade my wife to sit down and watch me reprise the break-in. (For which I used the variant mentioned by kwongshuchung, jetranger and grithog. Thanks to them and above all mathguy for such beauty.)

[@charleslivingston2256]

Wow. That was so enjoyable i had to watch it twice. Simon's reactions throughout are wonderful.

[@highpath4776]

I dont do these as such but I would have looked at Y - which must be at least 3 (y of two cells in same box). If x is 9 then look at the x-y which must be 6 and cannot be more than 6 . ( so 654321 ) and we know which line it is in in Box 7. The y squared box min is 1+2+3 = 6 + 10 (1 2 3 4 ) = 16 . so min of y is now 4 ( so x-y= 54321) can we make the y squared box = 25 36 49 64 81 ( well max is 9 8 7 = 24x2 48 + 6 = 54. . So that gives y now 4 / 5 / 6 / 7 x squared min is 22 so X must be 5 or more which is confimed as Min5 > Min4 . x cannot be 9 as the x squared box is 3(7+8+9) + 6 max = 78. x= 8 / 7 / 6 / 5 . and x-y = 4 / 3 / 2 / 1. This should give futher access to getting a first digit. ( Which Simons set theory break in is great at working the x-y value )

[@mathguy_1243]

My idea was hopefully not allow a break in by whittling down possible combinations and I think I left enough wiggle room where that isn’t possible. I had an alternative version of the puzzle that was wayyyy messier and it allowed for that alternative break in

[@LiquorStoreJon]

Happy Birthday, MathGuy_12!

[@longwaytotipperary]

Happy birthday!

[@mathguy_1243]

Thank you!!

[@hotwheelsnholdem4873]

20:00-22:00 I'll just take your word for it Simon

[@timcotton1782]

A standard axiom of subsequent squares is that X^2 = Y^2 + X + Y. Therefore, since that is true in this case, X=Y+1, or X-Y=1. That places 1 in R7C4 and R9C8, and makes the arrow in Box9 234=9.

[@map-reduce]

Oh wow, I remember that from maths decades ago, very clever! Would never have come up with it these days :)

[@Bonehead777]

I love this puzzle. Supremely clever. 🧐

[@mathguy_1243]

Thanks! :)

[@RWBHere]

Might be wrong here. I've stopped the video at about 15 minutes in. X=8 and Y=7. By elimination, smaller or larger values will not work in that grid.

[@RWBHere]

I'm humbled by your solving of this one so smoothly.

[@Mausemotte]

wouldve never gotten the set theory at the beginning but then i felt the 8-9 logic at the bottom was painfully slow i saw that right away. funny how simon has an easier time figuring out the more advanced logic and then glances over more simple logic

[@matteovisentin3692]

Craziest puzzle I seen you solve !! I’m impressed

[@wrexocane]

This is the most creative puzzle I've seen!

[@Awedish]

This puzzle was my introduction to SET hahahaha ... there were much easier ways to learn it

[@mathguy_1243]

😂😂😂 thank you again for testing!!!!

[@valeriovendetti7387]

10:09 On a sidenote, there actually is a minor constraint on the circled square in the upper left: it can’t be 2, 5 or 8 by using the divisibility by 3 of the renban lines together with the fact that the cage adds up to a square and 2 isn’t a quadratic residue mod 3

[@johnh2052]

Simon, sometimes Sven's app requires one to delete old sudoku data. In the past, I've seen that mess up the time. If I didn't restart the timer after the deletion, I'd always get a time of zero. That may be fixed now, but I'm not sure. (I haven't seen it for awhile as I always restart now.) I think the issue may also affect the number of solvers, and may be why you sometimes see yourself as the only solver. (And some times may because you _are_ the first solver, of course.)

[@jjockee1]

Does these rules only work for this specifik puzzle or is this for every sudoku-game? Like the circle adding up to the line etc

[@mathguy_1243]

Specific to this puzzle for basically every element

[@chrisburgess4959]

It is such a joy to see your mind in action - thank you for sharing your talents and enthusiasm!!!

[@beth7808]

Oh. I missed this originally and now the puzzle link is broken. I'm sad.

[@daviddejong3894]

Same about two weeks ago. But it's working again now!

[@stanleydodds9]

Here's a note on the first thing he thought about (but I haven't watched the video or tried the puzzle, so I don't know if it's relevant): The square numbers are always congruent to 0 or 1 modulo 3, i.e. they are never 2 modulo 3. You can check this by simply squaring all remainders modolu 3: 0^2 = 0, 1^2 = 1, 2^2 = 4 = 1 mod 3. So given that the two renban lines sum to a multiple of 3, and in total the cage is a squre number (Y^2), the remaining cell must be congruent to 0 or 1 modulo 3; it cannot be 2, 5, or 8. In practice, this is a lot easier to check by simply looking at the possibilities of the sum being 16 (1 more than 3*5), 25 (1 more than 3*8), 36 (0 more than 3*12), or 49 (1 more than 3*16). Then you can clearly see that the remainder is never 2. But in any case, modular arithmetic is a good introduction to mathematics beyond the usual numbers that we thing about, hence leading to group theory, rings, fields, etc.

[@LednacekZ]

7:10 he was literally one thought from cracking it. just continue the logic. unvind the arrows, add row and column 9 to one side and box 9 to the other and you end up with X^2=Y^2+X+Y. which translates to x=y+1

[@mathguy_1243]

Ikr!! So close! But I’m glad he got there in the end :)

[@nonagrey3422]

OMG you're Ren fans!!!! LETS GO!!!!

[@karsaanita]

I saw the squares and the difference and I knew I had to use x^2-y^2=(x+y)(x-y) somehow. Absolutely amazing puzzle!

[@titusadduxas]

2:47:21 - Wow! That was an amazing construction. It took me about 80 minutes to break in and I’m so chuffed I managed to do that unaided. I did have to backtrack a few times but I’m delighted to have completed it.

[@mathguy_1243]

Thank you so much for solving!

[@titusadduxas]

No thanks necessary by you Mathguy. The pleasure was all mine. How did you come up with the idea. I didn’t use the method Simon did. It was a painstaking elimination of digits until I ended up with 7&8 after about 80 minutes!! Good job I’m patient!

[@mathguy_1243]

Oh my you are very patient! Simon’s method was definitely my intended route, tho when I set the puzzle I used an altered p-ring instead of Aad’s set (which I found later) the idea behind the math is a concept I had been interested in since before high school when I wanted to develop my own way of approximating square roots in my head (nerd am I right 😂) I ended up needing to take the difference between consecutive squares a lot and found that pattern by accident

[@Aixler]

Figuring out the break-in was impressive

[@YellowBunny]

I solved the equation for X-Y in a slightly different way that I found easier to do in my head. Here's what I did: Y²+X+Y=X² Y²+Y=X²-X Y*(Y+1)=X*(X-1) and because the multiplicands are >0 here => X=Y+1 X-Y=1 And here's more or less what Simon did: Y²+X+Y=X² X+Y=X²-Y² X+Y=X²-XY+YX-Y² X+Y=(X+Y)*(X-Y) and because X+Y is not 0 here => 1=X-Y

[@bevstarrunner9472]

I feel like you are missing a step here.. the part which might help the lesser mathematicians understand it the most.. Y²+X+Y=X² (From the Sudoku - Sum of Green squares equals Sum of Blue squares) X+Y=X²-Y² (Rearranging) X+Y=X²-XY+YX-Y² (Adding XY and also subtracting XY i.e. net zero change, to add some dummy variables) ** X(X-Y)+Y(X-Y) (Factorizing X from first two terms and Y from last two terms) X+Y=(X+Y)*(X-Y) (Factorising (X-Y) from both terms) and because X+Y is not 0 here => 1=X-Y

[@Arcessitor]

@@bevstarrunner9472 I'll be honest, as a lesser mathematician, I'm still not understanding either of your two explanations.

[@kgbgb3663]

​@@Arcessitor Read "**" as "X+Y=". Presumably "**" is a variant of "ditto" or " '' " in some country.

[@barefootalien]

What a puzzle! I gotta be honest, though, I never would've broken into it. That geometric trick isn't something I've ever seen before, and the like, seven steps of logic in a row to get to it from scratch is... unlikely. But once Simon got the final equation, while still tricky, it wasn't too unapproachable! Was there another way to break in? Given how perfectly the arrows worked out for the substitutions, I'm guessing that's the intended (only?) path. Prior to that, I'd gotten that x-y = 1, and restricted y to 567 and x to 678, but without x² - y² = x + y, I couldn't get much further.

[@mathguy_1243]

Thank you! And no I don’t think there really is another way to make progress without at least getting the equation, however there are a few other comments that show how you can take a p-ring approach to the set instead of Aad :)

[@barefootalien]

@@mathguy_1243 I uh...

[@mathguy_1243]

No worries! It’s the most popular/recognizable use of set, CTC has done videos on it many times. The original founder was phistomafel and the end result makes a ring on the sudoku, hence the name p-ring. If you look up phistomafel ring there are probably many videos explaining it way better than I can in a comment section 😂

[@barefootalien]

@@mathguy_1243 Hmm, I've probably done them, then. I've done every puzzle in the Classic Sudoku app, and about 2/3 of Sandwich Sudoku. I actually wish variant sudoku made us use some of the more complex techniques more often. Most of them, the most advanced it gets is maybe the occasional X-wing. I think I've used one or two Y-wings, but could probably have used something else. My favorite advanced technique is... actually I dunno what it's called. The place I learned it called it "completing the square", but that also seems to refer to other things, and googling that name doesn't find anything on it.

[@shteevuk]

I solved this but didn't get anywhere close to the elegant solution Simon arrived at. My solution was a MESS. Reminds me of that time I solved a Phistomefel Ring puzzle before I knew what the Phistomefel Ring was ^_^

[@zirco77]

(for the less algebraically-inclined, there was a visual way of finding "X-Y" below) Wow! sets, maths, renban, arrows... such a wonderful mix (and I like maths and sets!). Thanks @mathguy_1243 for that hour of pure pleasure! Now, once we find out... ... ... ... with sets, that the difference of square is X+Y, then we can find out "visually" that X-Y=1 just by coloring cells in a grid. Let's say Y=3 (works with any positive integer). Color 3x3 cells. Now we need to color X+Y more cells to get X squared cells. By coloring Y more cells, we have a 3x4 rectangle. We still have to color X more cells so as to make it a X by X square. It has to be 4, one more than Y.

[@mathguy_1243]

Thank you and what an excellent explanation for the visualization!

[@jetranger57b]

I got to the break-in slightly easier than Simon did - I initially shaded all of the cages and arrow sums and noticed a pattern between the lower corner and some rows/columns. I then shaded only the lower 4 boxes (5,6,8,9) as four groups of 1-9, and contrasted them with columns 4 and 9 along with rows 4 and 9 as another four groups of 1-9, accounting for the fact that I was double counting two of the cells (4,4) and (9,9). After eliminating the overlap, and leaving the extras from 4,4 and 9.9 you can then switch to the equality comparison and eliminate the arrow sums - leaving the same position Simon ended up with - the x squared cage equaling the y squared cage plus the x and y cages.

[@mathguy_1243]

I’m so glad you found the alternate way of getting the set, it’s the same sort of method you would use for a p-ring: 4 boxes, 2 rows, and 2 columns!

[@grithog5399]

Same here. Although it took me quite awhile to get to it (I started by doing a bunch of max/min calculations), I thought this break-in was a bit simpler than Simon's.

[@FirstLast-gw5mg]

I think of this pattern as a sort of offset Phistomefel ring.

[@5t757]

35:19 for me, lovely lovely idea.

[@mathguy_1243]

Thank you!

[@MarushiaDark316]

Tomorrow may be Ren Day, but here at CtC, everyday is potentially REM Day.

[@solfeinberg437]

Once Simon added row 9 and column 9 to the green, the puzzle was sublime. Y was still in {4, 5, 6, 7} but X was suddenly one more. The geometry was perfect. Y^2 + X + Y was green and Y^2 was blue. They were equal. I was never going to add the whole bottom box 9 to blue just to get cell (7,7).

[@mathguy_1243]

Set can definitely be a bit counterintuitive, but Simon’s solve was nearly flawless, absolutely amazing to watch the way he works things out

[@mycroft64089]

My favorite detail of the puzzle is the only cage with a given value being a naked X + Y.

[@mathguy_1243]

You know, that totally wasn’t intentional 😂