The Perfect 6x6 Sudoku

[@dorlirahmeti7576]

Thank you so much for doing this puzzle. It was really a nice surprise after exhausting day I had today. The opening of this puzzle is probably one of my favorite I made. And the moment I thought of I knew I had to do it. And really glad you enjoyed it as well. Thank you so much for this and take care.

[@easternbrown]

Thanks for setting it! And the help from the title too.

[@zirco77]

Wow, such a exquisite break-in! (I won't spoil any detail here for whoever hasn't done the puzzle yet) The "somewhat well known" sudoku trick at its base makes for very interesting break-in by itself. Then your square root constraint on top of it was the icing on the cake! What a beautiful combination!

[@falloutfan2502]

Genius!

[@jasonmontgomery6122]

These puzzles help my mental health and give me something to work on to help with my anxiety. I'm sorry today was hard for you, and thanks for helping to lift me up some today!

[@chipsounder4633]

That was my GCSE maths coursework. Shouting at my past self now for not completing it.. 😂

[@martysears]

What a well deserved feature. Dorlir told me he was making a puzzle based around a Pythagorean triple, but I hadn't solved it yet. Even knowing that vital information when watching this video, I still couldn't spot how it would be used, even though in hindsight, the required SET is staring you in the face. Simon's ability to intuit and then quickly use these things in the correct way is his real special talent that makes him stand out as a solver. In the last two years I've watched a lot of streamers solving puzzles, and some of them are absolutely brilliant in their own ways, but I still maintain that no one has a better 'spidey sense' intuition than Simon to get inside the head of a setter and find the correct intended logic. Great video, excellent puzzle, lovely way to spend half an hour on a Sunday evening

[@MasterHigure]

His ability to ask the right question at the right time is just marvelous. I can always follow Mark and Simon's logic, and the featured puzzles always seem like I could solve them myself, the way they present them. Until I'm stuck staring at a grid for an hour not knowing where to go next. Then I realize the puzzles are actually really difficult, it's just Mark and Simon making them look easy.

[@mattjones2477]

Thanks so, so much for the birthday shout-out for Isobel, Simon. Rest assured she had multiple slices of chocolate cake with plenty of icing. She’ll be overjoyed to see this video in the morning - the downside of being 9 is that these videos are released post-bedtime!

[@floydmaseda]

This feels like another puzzle that should be featured on Numberphile!

[@janerobson2297]

Yes, @numberphile - Brady please note

[@AaronSmith-i8y]

I'm not sure I have ever used SET in a 6x6 before. What a great twist! I got stuck, started reading Marty Sears comment, went back and solved.

[@alexgordon7312]

You are correct about the triple but you could have proved it by testing cases of your options as well, and noticing that 3 and 4 were the only orange pair that worked! :) Great puzzle and a joy to watch the solve

[@harshitsingh3459]

Opened the sudoku, realised it's impossible. Then started the video, heard 'set', realised along with Simon about Pythagorean triplets, went back and solved in 5:37😅

[@emilywilliams3237]

What a wonderful solve and video, Simon. The puzzle is fascinating, and your delight when you recognized the "trick" (if I may use the expression) was fantastic. Thanks for the video!

[@ingiford175]

Some Triangle information: 3,4,5 is the only Pythagorean triple that all three are are single digits. The next 'smallest' primitive is 5, 12, 13, and though there exists another one between the 3,4,5 and the 5, 12, 13, but that is simply the double of the first one 6, 8, 10

[@sniper1a259]

Fun fact, a 3 4 5 triangle is known as Egyptian Triangle because it was used in Ancient Egypt to create right angles

[@darreljones8645]

Yes! And the term "primitive triple" is the one Simon was searching for in this video. Even if you allow for the possibility that the two smaller numbers are the same, you'll never get an integral longer side because of the irrationality of the square root of 2.

[@chipsounder4633]

It's amazing that this puzzle had elements of what was my maths coursework at school. Fantastic representation 🎉

[@A_CC_K]

Ooo a Dorlir puzzle, best way to spend an evening.

[@aragaer]

If you look at the digits filled in last (row 5, columns 1, 2 and 3) they are the "3-arrow" adding up to 12. Which means that the rest of the green area adds up to 13. Which limits the r5c4 to either 1 or 2 and another of those two goes onto the arrow. Now the minimum you can have in the upper half of the green area is 11 and if you replace 5 with either 7 or 8 you get too much, so it must be 11 and that means r5c4 is 2 and the arrow is 1+8.

[@TPH250290]

Unbelievable. Possibly the first time I've spotted SET within a couple of minutes, and then after that it was a really elegant 11 minute solve.

[@TPH250290]

Jeez, just watched the rest of the video. Why did Simon explain it by waffling about hypotenuses when you could just say that 3 squared is 9, 4 squared is 16, and if you add those together you get 5 squared which is 25. And that if you made R5C5 a 6 or a 7, there's no way of making other squares add up to 36 or 49.

[@penningmeestercgkdelft9159]

Wow! I absolutely love Pythagorean triples and once when I was young found a simple method myself to generate them all, but to find them back here in a sudoku with set theory, is just stunning and surprising. Take a bow! Simon, you are absolutely right: (5,12,13) is the second simplest triple behind (3,4,5). You can derive them as follows: Let x and y be two natural integers, and require that x>0, y>0 and x>y strictly. Next, generate three other integers: a = x^2-y^2, b = 2xy and c = x^2+y^2. Then you will always find that a^2+b^2=c^2, so (a,b,c) is a Pythagorean triple. The simplest case is (x,y) = (2,1) which yields (a,b,c)=(3,4,5). Next, one can find (x,y)=(3,1) which yields (8,6,10), and (x,y)=(3,2) which yields (5,12,13), and so on. In fact, any rational fraction y/x will have a Pythagorean triple associated with it, although many have common factors so they are not all unique. Furthermore, we have arctan(b/a) = 2*arctan(y/x), which is also a nice property. As (3,4,5) is the only triple with only sudoku digits, that's obviously what has been used here 🙂

[@HarleyMaebry]

4:39 “[Throwing a perfect game in 10-pin bowling] strikes me as an incredible achievement.” Best unnoticed, unintended pun!

[@eddieharwood7788]

Wow what a fantastic bit of intuition Simon. Once I got the idea at the set I went on to solve the puzzle without too much difficulty, but still not smoothly.

[@olivier2553]

My own explanation about multiplying a 3,4,5 triangle is that I do a change of units. Instead of a triangle of 3, 4 and 5 meters, I will use a triangle of 3, 4, and 5 wa (A length unit in Thai language that represent 2 meters). It is still a 3, 4, 5 triangle, Pythagoras still applies and now if we measure in meters it is 6, 8 and 10(or 6000, 8000 and 10000 in millimeters). Great idea by Dorlir and great solve Simon.

[@mscha]

Wow! I would never ever have found the breakthrough all by myself. Only when Simon uttered the magic words (that I won't repeat here, spoilers) and pointed at the relevant rows, I figured it out. After that, it went smoothly.

[@Mujaki]

What would you call this new piece of grid furniture? I'd suggest "Squarrow."

[@sniper1a259]

Pythagorean theorem was not something I would expect in a sudoku puzzle Simply beautiful

[@Gonzalo_Garcia_]

18:18 for me. Fantastic puzzle!! It took me a while to spot the break-in, although it does look quite obvious once you see it. Really enjoyed it!

[@levb258]

I loved this puzzle. Realising what the break-in was was amazing! It's really nice and simple but so beautiful

[@bobblebardsley]

I'm old enough to remember the days before widescreen TVs and I wasn't an interesting child so I used to be very happy about the fact that you could very easily calculate the dimensions of a 4:3 TV screen knowing only its diagonal measurement (which is what they put on the box when you buy a TV). Then they gave widescreen an aspect ratio of 16:9 and I'm still annoyed. If only they'd made it 15:8 instead...

[@ericpraline1302]

A very satisfying solve with the added benefit that I can feel smug for a while for having worked it out.

[@jonhansen9622]

"A perfect game in 10 pin bowling? That STRIKES me as an incredible achievement." Excellent pun, indeed

[@xerodeus2337]

I'm very proud I saw the set theory opening! I love that I'm able to look for those when I don't see an obviously easy opening

[@mstmar]

i struggled pretty bad finding the break in. i saw that every digit on the bottom arrow ended up on the longest arrow by sudoku. that meant that 3 of the digits on there added to either 6, 12 or 20. i spent most of my solve time trying to figure out which if any of those broke the puzzle with no luck. it did give me an alternative to the purple squares though once i got some digits. starting from 27:45, the bottom arrow's digit add to 12 and are on the green arrow in box 5. the top of the green arrow is at least 11 (1,2,3,5) and 1+ for r5c4. these sum to at least 24, so the green arrow only has 1 degree of freedom meaning there's a 1,2 and 3 on the top of the green arrow plus a 5 or 6 and a 1 or 2 in r5c4. we can't add a 6 in the mix or we'd have 7 digits (1236 on green arrow, 45 in diamonds and the 7 or 8 on the orange arrow).

[@sacredsock8031]

Some one needs to drag Matt Parker over here, I was just reading of his love of triangles

[@adrianp5465]

This was a brilliant puzzle to solve. After a bit I felt like giving up on it and just watching the video, but after years of watching the channel, I knew I had the ability to solve it if I just kept chipping away at it. Thanks Simon and Mark :)

[@nathanmays7926]

Sub 1-hour? That means I get to try it myself! brb...

[@zhoppa]

that's what I thought, and was completely embarrased!

[@Nerdnumberone]

​@@zhoppa It's a rather unusual break-in, despite being hinted.

[@denis1996DV]

Let us know how it went 😂

[@piarittersporn]

After a big mistake in box 6, which I didn't recognize for a long time, I finally found the solution and will now take a relaxed look at Simon's solution. I'm sure his is the smarter one.

[@chocolateboy300]

I finished in 128 minutes. After watching the video, it seems obvious that I missed something big and of course it was SET Theory. I was wondering why the puzzle was so hard, but I got lost in the sauce a little and kept doing calculations. I saw that there was a limitation of the row sums and the column sums in the bottom right diamonds. That seems to line up with the SET Theory. It is like I manually performed the calculations of the set, rather than actually proving and using SET Theory. I don't know why my brain ignores it so often. I feel bad for the puzzle for what I did to and I feel like I have to apologize to it. I apologize puzzle and Dorlir for what I did to you. You deserved to have a better solve. Nevertheless, once I did figure out that 345 were the numbers, the puzzle became a lot more fun. Great Puzzle!

[@_inabox]

this one reminded me of the difference of squares puzzle from few years ago. this one was bit easier but both were brilliant

[@Paolo_De_Leva]

Beautiful idea by Dorlir and brilliant solve by Simon. 👏👏👏👏👏👏👏👏👏👏

[@jonathancard4466]

When I was young, a perfect 300 game in bowling was rare enough outside the pro tour, there was often a list on the wall of everyone who’d ever done it at that facility.

[@GraemeMcRae]

I put the square root in the diamond, the diamond in the coat, and the coat on her!

[@isaacthek]

This was quite fun to work out what the possible options are based on the limitations of the squares and the assembly of 6 digits...

[@xerodeus2337]

48 Minute solve, and a very very beautiful puzzle!

[@letsmakeit110]

was just about ready to give up on variant sudoku since i hadn't been able to solve anything on this channel in 6 months. somehow managed this one on my own, including the break-in.

[@nicka3697]

I got down to the fact it was 12345 and 7 or 8 but had to do the entire puzzle with 7/8s and 1/2s and colour the 1/2 pairs to solve it. It took me a bit longer than Simon without wishing anyone a happy birthday!

[@Forbizz]

“I don’t know enough about maths” says man who has been featured in a numberphile video

[@johncox7169]

I did this one a completely different path than Simon. After realizing that the 3 sums were 3, 4 and 5, I realized that whether it was a 1/8 or a 2/7 pair, the 3 (9?) arrow always added up to 12 (3 + 3 squared.) And due to the nature of 6x6 Sudoku, that mean that the top row of box 5 also added up to 12 (of the 25 we are allowed.) So then I saw that the first 4 digits of row 4 + r5c4 would add up to 13. If I minimized the first 4 digits of row 4 (1+2+3+5) they added up to 11 (and I knew that 6 wasn't a digit in this puzzle because of the 9 arrow), meaning that r5c4 had to be a 2 to make the 13 total I needed to complete the 25 arrow. That also told me that the 9 arrow was a 1/8 pair as 2 was no longer available in box 6. The rest was just Sudoku.

[@JohnGottschalk]

Did the labeling just fine, but was blanking on comparing the rows and columns, after that insight, managed to figure the rest out in a total of 27 minutes.

[@MorganZex]

Unlike the last filomino puzzle which was intimidating, had a 5 stars difficulty rating but gave me no troubles, this little elegant 6x6 was absolutely impossible for me. I could have solved it by trial and error but that's no fun. I enjoyed watching the solve and had to admit that yes, this is out of my reptile-brain league.

[@Xiuhtec]

Once again I'm hopeless at finding sets. (I understand how they work, just not how to look at a grid and have that a ha moment and start marking them off.) I stared at this for about 40 minutes without finding a break-in after starting the same way (marking possibilities in the diamonds).

[@Dalixam]

Uhh, I love these smaller sudokus when they're well made!

[@ConsciousAtoms]

There's an even amount of odd numbers in a Pythagorean triple. With 0 odd digits the only possible combination is 2, 4, 6, which is not Pythagorean. With 2 odd digits, one of the odd digits must be in the "a^2 + b^2" part (the orange part of your puzzle), the other odd digit must be in the c^2 part (the green part of the puzzle). Now, the green part digit is one of 567, and since it must be odd it must be 5 or 7. If it is 7, the orange part must sum to 49, but with the maximum numbers (4 and 6) the sum is 50, so a sum that is just one less than that is impossible. Therefore the green digit must be 5, for a sum of 25. Now both orange digits must be less than 5, hence the "456" number must be 4, leaving 3 for the "234" number. In other words: the 3, 4, 5 triple is the only possibility.

[@jorghimvenerablebosstein7568]

Once you know what the 6 digits are, it’s really easy from there. I skipped ahead in the vid where Simon figured out the 6 digits and solved it from there. It took me only 6:26. 😎🔥

I find it "cute" that 22+23=45. Great puzzle.

[@77kaczka77]

Beautiful

[@michaelkelly8955]

Playing golf in a gale and the only downside Simon had was, it played havoc with his hair. I think it was George Bernard Shaw who said, “golf is a way to ruin a good walk”.

[@Antiknight]

00:09:11 cute puzzle with a nice idea!

[@donaldsnyder1543]

There is most likely an easier way but as long as you do the same thing to both sides of an equation it remains equal. So you could just do an exhaustive check by going : (a^2 + b^2) + 1 = c^2 +1 , etc until you get there. Then you'd move on to multiplying both sides if addition yielded nothing. I'm not enough of a math nerd to know if this approach would be necessary or not tho.

[@hom-sha-bom]

Bro you don't have to have every Pythagorean triple memorized to solve this puzzle. a²+b²=c². You already know c is 5,6,or 7. 7² is 49. Do any of the a² or b² candidates sum to 49? No. Do any of them sum to 36? No. Do any of them sum to 25? Yes. 4² + 3² = 5².

[@diederikvandedijk]

It's not too hard to test the possible combinations of squares added up: [2, 3, 4]² + [4, 5, 6]² = [5, 6, 7]² » [4, 9, 16] + [16, 25, 36] = [25, 36, 49]. 4 adds up to 20, 29 or 40, so it doesn't work. 16 adds up to 32, 41 or 52, so that doesn't work. 9 adds up to 25, 34 or 45, so only the first option works: 9+16=25 » 3²+4²=5² indeed.

[@Rukimix]

I had a different break-in but resulted in the same digits being found, (after the 3/4/5), I knew that 3 and a 7/8 were in r5 of box 5, which meant 1 or 2 had to be in r5c4 and 4 had to be in r5c6, not as elegant but still worked

[@Ardalambdion]

I did Thank You Very Much by James Sinclair, the weird construction 11x11 puzzle that you recommended in your email. Without support at all.

[@deltaangelfire]

23:44 I mean you don't have to disprove all the pythagorean triples, just the ones you could possibly make. starting with 49 (or 7^2) subtract 36, is the remainder (13) a square? no. subtract 25, is the remainder (24) a square? no. subtract 16, is the remainder (33) a square? no. then eliminate 6 the same way and the only thing the diamond in row 5 could possibly be is a 5.

[@LarkyLuna]

For an irreducible Pythagorean triple one of A or B is even. If both are odd, C² has a single factor of 2 on it and C is not an integer Let's say B, and A and B are relative primes. I'm cheating a bit but let's say B is 2pq with p >= q => 1 Then B² = C² - A² by Pythagoras And 4p²q² = (C+A)(C-A) or p²q² = ((C+A)/2)* ((C-A)/2) Since everything is coprime: C+A = 2p² C-A = 2q² Solving the system: C = p² + q² A = p² - q² B = 2pq This allows you to generate any triple you want given p and q Since C < 10 in the puzzle, p and q < 3 Since A > 0, the only solution is p = 2, q = 1 A = 2² - 1² = 3 B = 2*2*1 = 4 C = 2² + 1² = 5

[@LarkyLuna]

It's not that hard to actually check in the puzzle which values of squares would actually work with the penciled digits and be left with 3 4 5, but here's some number theory anyway

[@bobblebardsley]

I walked across my local golf course today and was ankle-deep in water at times so I have no idea how you managed to play an actual round 😄 Also as a Robert I'd like to say I wholeheartedly do not endorse Hurricane Bert's shenanigans. Naughty Bert.

[@SirBradiator]

A perfect game of tenpin bowling is not rare at the professional level, it happens regularly in competitions. But still a fantastic achievement

[@pancentricism]

I can see you expounding this one on Numberphile

[@michaels4340]

I got lost because I didn't think of finding the Secret, but I enjoyed watching the video :) and the UK pronunciation of "Pythagorean" (which is probably more correct than the US version, but never mind :P )

[@Kimosabe4667]

The biggest difference between the UK and US is our common language: p-eye-thagorean vs pith-agorean!

[@NB_703]

I’m confused how you can do set theory with this puzzle since it could be populated by digits 1-9. Wait. Is it the same 6 digits for the whole puzzle?

[@gordonglenn2089]

A Pythagorean Triple puzzle following 11/23 (US style) Fibonacci Day! 😍

[@perigin3]

Damn i did absolutely none of that with the intended break in, just used a lot of arithmetic on the r6c4/5/6 combo needing to appear on the line in box 5 (and some examining of the various permutations of 16 on the top right arrow)

[@mxpxorsist]

what a nice puzzle 12:42 for me

[@LednacekZ]

I got the set theory, but completely missed the Pythagoras.

[@srwapo]

44:21 after looking at the video. I got the set theory, but kept breaking the sudoku. For some reason, my brain kept dealing with the diamonds as averages instead of square roots.

[@emdiar6588]

As usual I battered my way to the opening, using shameful bifurcation. I got there in the end but seeing the 'set' method now, I am face-palming so that it hurts. Of COURSE it was elegant - it's Dorlir after all.

[@grahamania]

00:36:20 for me. Not a quick time, but a great puzzle! Kind comment.

[@AditaBattleForge]

oh my gosh... tried this on my own, but did not spot the set. Was able to rule out some of the squares but didn't come far. Watched to minute 20 of the video and was dumbfounded. Finished the puzzle in less than 5 minutes from there.

[@Thegeektoendallgeeks]

00:17:57 i listed the options for the diamond, then i got stuck and proceeded to read the title and the pythagorean triple became clear and the rest fell in to place pretty quickly

[@psymar]

16:45 for the deduction I'd never have found

[@DonaldSjervenE]

4 squared plus 5 squared is 41. 9 squared is 81. QED.

[@robofox97]

yknow ive wondered for a while now, been watching for about a year, maybe two i havent kept track, but whats the origin of the '3 in the corner'?

[@michaelkelly8955]

I once got 2 on a maths exam, I think I would have gotten a 5, but I spelt my name wrong.

[@marcoballan4067]

He almost made italy's flag🇮🇹

[@williambond6340]

#21 solves shown in video, #1664 five hours later

[@joekerr3638]

Fuzzy logic

[@ThePolymathlete]

I've never heard Pythagorean pronounced like that

[@ichselbst4798]

Sometimes I totally do not understand Simon´s way to think... Though he had the base of solution at minute 19 (which I´d never get by my own) he startetd complicating his way of thinking ending in several minutes of gibberishing

[@HunterJE]

12:48 Long-winded way to rule out this minimal arrangement-123 in row 2 means the square in r2c2 must be a 4. But a 16 in two cells must be 79, at which point the six digits in play must be 123479. But that means r5c5 must be 7, since 5 and 6 are out. But a 7 clue which needs a 49 sum, and the maximum fill for that clue's line using the available digits is now two sets of 3+4+7+9, which only come to 46!

[@HunterJE]

(That said the actual set theoretic solve path gets the same result for free and is both more pleasing and more broadly useful)

[@thescrewfly]

Peculiar. I've always pronounced Pythagorean with the emphasis on the "thag" syllable (I'm British) and pronounced the 'e' as "ee". I have never heard anyone emphasise the "gor" before. Does anyone know if this Is an American thing? I know Simon is prone to using the occasional Americanism, but this sounds very odd indeed.

[@johnzikis4811]

@@thescrewfly American here and I have never heard it pronounced this way

[@michaellautermilch9185]

33:40 hate speech warning

[@michaelandersen-kk4fc]

another successful solve by divinity. as I don´t believe in Divinity this channel is clearly fake. Month and years of perfection is simply impossible what is the point of watching somebody that never fails or make mistakes. good for you Simon manipulating the results so you look like the perfect solver. all hail the god SImon, perfect divinity

[@letsmakeit110]

ctc doesn't upload failed solves. its not deceptive the channel is open about this. just doesn't make for a good video and it doesn't help if ppl want to try on their own and get stuck.

[@Kraxxus-vonDalaran]

en.wikipedia.org/wiki/Pythagorean_triple