Embarrassed to see I'd left a display bug on my solver and it's now immortalised. The row highlight on a found strategy. All fixed now. It inspires me to add a permutation option on the solver. A very interesting technique well worth knowing about.

@MyReligionIs2DoGood

3 жыл бұрын

Love your solver, Mr. Stuart! Not sure if you already did, but could you make it possible to track back more steps than one, please? It doesn't let me go back more than one - or am I just too blind to find the option?

Since you seem so curious, if you'd like to go down the rabbit hole in a non-sudoku setting, this is exactly an application of reduction (the term you actually used in the video) in computer science. One of the more famous applications is in NP-completeness, which requires a bunch of background, but is pretty interesting to get into. Suppose you have a map of countries bordering other countries. A common way to represent this in computer science is what's called a graph, where each country is a dot (vertex) and each border between countries is a line (edge) connecting those two dots. Suppose you want to colour the map with only three colours such that no countries bordering each other share the same colour. This is a well-known problem for graphs called colouring. For a graph, this means colouring each vertex such that no edge connects two vertices of the same colour. Rather than putting effort into colouring maps, researchers have put effort into colouring the more abstract graphs. Any solution that applies to the graph can be easily applied to the map. You could say we've reduced the problem to something more well-studied. As a more practical example, suppose you have 10 delivery cars and a predetermined set of deliveries where you know how long it will take to deliver to that address and return to HQ. You can have many different deliveries overlapping in various ways. How do you allocate cars most efficiently? How many cars would you need to do it perfectly before adding more cars becomes pointless? Both of these can be answered by making a graph where each delivery is a vertex and overlapping deliveries are connected with an edge. From there, you can colour the graph optimally, which is a very well-known problem with lots of research, and then apply the results back to your delivery service. The number of colours you need is the number of cars you need to make it perfect. Being efficient with 10 cars means colouring the graph as optimally as possible with 10 colours. We've reduced the delivery system to well-known graph problems. There are a *ton* of things that can be turned into graphs, making them an extremely useful abstraction for real-world problems. You can also turn sudokus into graphs where each cell is a vertex and each edge connects two cells in the same box, row, or column. Each number would be a colour. This comes remarkably close to the technique discussed here. The point is, you can reduce sudokus in general to problems in graph theory. You can go further the other way as well. There's a famous problem called the Boolean satisfiability problem (SAT). If you have a Boolean logic expression such as "(x AND NOT y) OR (z AND x)", the problem asks whether there is a set of true/false values for x, y, and z (or whatever other literals you might have) that makes the overall expression true. You don't have to know what that solution is, only whether it exists. It turns out that we know how to reduce this to a more specialized version called 3-SAT where you have a general structure "_ AND _ AND _ AND ..." and each _ contains 3 literals maximum and only OR and NOT. We know how to convert any expression to this form and 3-SAT tends to be easier to work with, so we can do work on 3-SAT and apply it to all expressions. Where NP-completeness comes in is that it turns out we can reduce that graph colouring problem from earlier (whether you can colour a graph with 3 colours, note: not giving an actual solution) to 3-SAT. We can also reduce 3-SAT to that graph colouring problem. That means any progress we make on either of these problems applies to the other one. What's more is that there is a whole slew of hard problems that are also in this set of problems that reduce to each other. If we figure one out really well, we can do the same for all of them. You might have heard of the P=NP problem, which has a $1 million prize. In short, to solve it, someone needs to either prove that any of these problems can be solved "fast" (for a precise definition of fast), in which case we can apply that to the others, or that any of them cannot be, in which case none of them can. That got long, but if it piques your interest, know that there's a whole bunch more and you know how to get started on looking for it.

@ahnowrong6457

4 жыл бұрын

how do I save a comment

@Qazqi

4 жыл бұрын

@@ahnowrong6457 Haha, you could always copy it to a text file. You can even bookmark a text file on your computer if you want through `file://` URIs, or put it in Pastebin, or an email to yourself, or whatever else works best for you.

@DemonLilyYT

4 жыл бұрын

Now you just need to make sure that SETH our lord and savior is real, protecting us from insane speed at solving them parametrized NP-complete problems.

@johnharriman85

2 жыл бұрын

One of my first professors in university was Kenneth Appel, he of the '4 colors suffice' fame! very interesting read. You may ruin the internet with such intelligence if you're not careful.

@dickybannister5192

11 ай бұрын

think you're under selling this, and I'm not sure "reduction" covers it either. this is actually pretty shocking. a random selection usually assumes a bifurcation: i.e. you have NO method to decide which might be right. so you might as well randomly try one. but, if that fails, you have to "go back" and try the others till you get the right one. one other possible use is to crank probability theory to reduce the likelihood of getting the wrong answer. this doesnt guarantee the correct answer, but with successive trials makes it less likely in a large way. these Random Algorithms are useful because checking any solution is right is easy, so the trade off is productive. if, here, you define a strategy as a linear collection of tactics (i.e. the list of things the computer might try), you can find an optimal solution in terms of the "best strategy" for a given sudoku instance in terms of the number and complicatedness of the tactics used. note: a strategy MUST work, be finite and run in finite time. again, random selection is never usually optimal because it is the worst possible! you have to try everything. note: "finding the solution" and then just saying "do" the solution... isnt a strategy... you havent explained WHY it is right: you have to DESCRIBE how to find it. this isnt a "proof" because we cant say, for sure, that there is a shorter tactic we havent discovered yet, but this is, frankly, bonkers. it basically says that the best strategy involves choosing at random. this is the first time that (albeit with backtracking, ie. actually undoing some part of the solution you know to be correct) that i've come across wherein random selection can actually be shown to be part of the best known strategy.

Next Episode: How to Transform an Easy Sudoku into an Extreme Puzzle.

@patricgmuer2988

4 жыл бұрын

nmorawski22 - Random Stuff let me guess, to change from easy to extreme you delete some numbers

@sircheesethethird6179

4 жыл бұрын

oh, i do that all the time. it’s called, *overthink the living daylights of yourself, the sudoku, and everyone within a ten mile radius.*

@monolyth421

4 жыл бұрын

It involves barbed wire and high voltages

@Freeman4815

4 жыл бұрын

Close your eyes

@thereoncewasalimerickwriter

3 жыл бұрын

I can already do that.

At 17:13 when you say "huge number," of course, that number is just 6.

@letMeSayThatInIrish

4 жыл бұрын

Well, in general it's n! which is huge!

@jlhidalgo

4 жыл бұрын

I think the "huge number" he was referring to was the number of potential placements of nine sevens, nine eighths and nine nines on the 27 cells available for them, which applying simple combinatorics gives you exactly... a huge number, I guess, since I don't remember my combinatorics and I'm too lazy to look that up 😅!!

@Gamesaucer

4 жыл бұрын

Well yes, but the _valid_ potential placements number to 6. And he was talking about valid solutions only.

I like this technique, which seems obvious in retrospect. As I commented on the earlier video, I used colours for three cells (the bottom row in my case) and then used normal Sudoku rules on them. It's not possible to add notes to cells, admittedly, but that's not much of an issue with only three colours. The bonus is that once one of the existing 7,8,9 is coloured, you can go back and fill in the rest without having to relabel anything.

@Lucasaraujo-um1pr

4 жыл бұрын

i did that too!!!

@Hichamnostik_YT

4 жыл бұрын

nice idea!!!

@teambellavsteamalice

2 жыл бұрын

So cool, I took your idea and expanded upon it to make this trick helpful in many more situations: So you keep the 789 where they are, just work your way onwards with a yellow, green and blue label until any of them resolve into a 7, 8 or 9. If it doesn't help wipe them and start again with another set of three squares... OR ... add another set of three colors! Add orange, red and purple and continue with these temporary labels until any resolves into EITHER 7, 8 or 9 OR into yellow, green or blue!

@DeborahJLeslie

2 жыл бұрын

@@teambellavsteamalice I wish I could see the work in progress. Do you color the given or derived 7, 8, 9 in the central column? How do you begin with the colors? Is your technique a colorful way of bifurcating?

@teambellavsteamalice

2 жыл бұрын

@@DeborahJLeslie I think bifurcating is a proper way to describe it. It's like have unknown a, b and c variables that can be 7, 8 or 9. The given 7, 8 and 9 will only get a color once they match with one. I think it's equally fast as you only advance from the color side. In another sudoku, there may be more to work with on one of the numbers and you could branch from both sides until a number matches a color.

Hi, 1. This is essentially Mark's favorite method of bifurcation, in a more graphically palatable way, which is specifically tailored specifically for the situation of a plethora of most number with a dearth of a few others, with the benefit of telling you exactly which fork is the correct one with the benefit of fore-hindsight. 2. If you replace 7, 8 and 9 with stars, you'll get battle sudoku... 3. Which brings me the idea of, instead of confusing the issue of relabeling the numbers, to use symbols (say a star, an orange and a banana) or just letters (A, B and C) to potentially represent either of the 3 numbers (without deciding which), starting from another square, row or column by populating them randomly, and only once you populate the central column with them to accordingly ascribe a certain value to each symbol. This has the benefit of not only being far less confusing, preventing errors in the last stage of conversion and sparing the use of color-coding to anyone using pen and paper, but will also save you some work and time, because the moment you manage to populate 2 out of the 3 original positions with symbols or letters , you can decode their meaning, translate them into numbers in the cells you've filled with them, and continue solving with regular numbers as normally. 4. Which only begs the question of where to start. Now, you've mentioned the bottom-right square seemed like a suitable candidate. And were proven to be correct. But you've never explained what makes it - or any other - suitable!

I think this is a version of a trick I use some times, using pencil and paper, but mine is (maybe) a bit more applicable and a bit less "intrusive": whenever you have two or more digits of which there are only one or zero (there can only be one digit not present in the puzzle, of course), and you see some cells where some of these numbers should go, and it seems that solving those cells would help advance the puzzle, start using "a" and "b" as new numbers, or "a", "b" and "c" as on this case would be required, and follow the normal sudoku techniques with those additional "numbers", as if the single cell with each of those numbers was empty. For some time you'll have cells in the puzzle with small numbers like "23a" or "pairs of 7 and a", or things like that. But whether you get to "solve" one the original cells that you removed with "a", "b" or "c" (which are their only options, by the way) you know the equivalence between that letter and the original number on the cell, so you replace that letter everywhere and keep on with the sudoku normally. It's like solving the X on an equation, that's my mental image for this trick. Sorry that my explanation is awfully, yet I hope it's somehow understandable!

@wizzolo

4 жыл бұрын

yes it's basically the same trick, with pen and paper you can assign letters, or arbitrary symbols ( X, O + etc ) to a set of empty cells, and solve by those, then when you solve the cells with the initially given digits. you translate them back. It would be nice if the sudoku software allowed extra notation letters/symbols for this purpose, but i guess the use would be relegated to this particular trick, and very little else.

@johnharriman85

2 жыл бұрын

I wrote my long winded version of this above before I read your comment. Cheers.

What a wonderful technique! It makes me happy to see - and it`s actually very simple. Like changing the 7-8-9 with x-y-z and then putting them back afterwards. Beautiful !

The label at the beginning (Extreme, Diabolical, etc.) shows the type of techniques the solver needed to complete the puzzle. The score is a cumulative running total, where every technique on the right is worth a different number of "points." So, Extreme-183 means it required some Extreme techniques to solve, but not very many of them. Extreme-800 would mean it not only required Extreme techniques, but it also required very, very many of them. It's theoretically possible for a puzzle to be "tough" 200+ if it's requires many, many applications of "tough" level techniques. Also, when the solver finds a bunch of naked singles or something in a single step, it only counts it as "one" technique, whereas if a naked single reveals another, it adds points for that twice. This puzzle "lost" a whole bunch of points compared to your average puzzle because of that, and because starting with a lot of givens meant it was shorter than average as well.

After you highlight all the squares of one digit to colour them, you can also replace the number with one keypress as long as you keep them highlighted.

@thorndeux

4 жыл бұрын

Haha, yeah that drove me nuts as well :D

3:44 When a Sudoku champion says "What the heck is that" then every average person should just give up on this puzzle.

To change the numbers at the end a way faster method is to delete a number first, then put the deleted number in in correct place and then keep doing this with the last swapped number. The last one will fill the blank spaces

In second year integral calculus, we joked about "assault with a deadly substitution". I waited 38 years to use that line in real life. :)

I got it right up until all the 7-8-9. I am kinda proud of myself (and I found it rather easy) but then omg I looked and looked and could not figure it out 😅

This technique is fascinating ! Thanks for the demonstration

This puzzle was a great way to show off this technique that made the puzzle incredibly simple. I attempted the puzzle before watching the video and came to almost the exact same strategy for solving the puzzle. Instead of assuming the locations of the 7, 8, and 9, I colored those 3 boxes to label them, and then solved the puzzle using those colors as a fill- in until I knew what they had to be, which was when I had basically solved the puzzle. I love hand made puzzles like this because the solutions are almost always elegant in some way.

Great video as always, it is a joy to use your excellent website of interesting puzzles, and your series on new sudoku techniques is amazing! Thanks! With the well deserved praise out of the way, I got a smile out of this: "... a fixing out of the HUGE number of permutations ..." One handed counting: 1, 2, 3, 4, 5, huge, huge, huge, huge, ...

Found this from the greatest hits book and wow. This is so fascinating. This is actually a really nice application of the math field of group theory. I remember in my class on it we were able to take two seemingly completely different things and show that, when you look at the structure of it and ignore the details, they’re actually the exact same thing. Or “isomorphic” if you prefer math jargon. So like in this case, individual numbers don’t matter. The structure of the puzzle is what matters. You could replace 1-9 with A-I or even 9 pictures of cats and it the solutions would all be identical, in a way. So why not just swap the numbers around?

Hi, Simon. Thanks for your great job. Funny. After I started this video, I as usually clicked on the puzzle page to try it myself. When I finished the easy part, I used the technique I invented myself (or re-invented, most probably). I call it 'traffic light'. I assigned colors: red, yellow and green to digits 7, 8 & 9 and started coloring the grid. At some point I was able to disambiguate 3's and 4's and completed coloring. The last step was very easy: to assign each color to a particular digit by noticing which colors match the open 7, 8 and 9. The whole task took me just several minutes and I was very proud of myself ...just before I started to watch the video and realized that your method is absolutely the same, just executed in a different way. I still believe that coloring of the grid is easier and more stylish than putting arbitrary digits and I hope you use 'my' method in a future video. Sorry for my English - I am not a native English speaker.

@craftypam9992

4 жыл бұрын

No need to apologize for your English - I don't think I could have expressed myself that clearly!

Simon, a great addition to this video would have been to see if removing the 7-8-9 from the middle column resulted in the computer solver showing that there were 6 possible solutions to that emptied puzzle (i.e. the number of possible permutations of the 3 numbers). Fixing the order in the middle column would therefore be shown to be choosing 1 particular solution.

What an incredible idea - and so simple. But if I were solving this on paper, rather than on an app, instead of putting 7, 8 and 9 into the last box, I'd use A, B and C. Doing the final swap back to numbers would then be less confusing.

I solved it without the technique you´ve used... but it took me like 50 min. Seeing you solve it this easily puts a smile on my face

@xian81_33

4 жыл бұрын

Impressive. How did you achieve that? Did you apply the strategy online solver did?

Unlike Ken Ken for example, and many variations on Sudoku, in classic Sudoku, the digits 1 - 9 are merely labels themselves and can be replaced by the letters 'A' through 'I' for example, or the nine members of the 2016 Chicago Cubs starting lineup, or 9 distinct colors (colours if you will). So, maybe it would be more intuitive to color the 3 empty squares in box 9 and to consider the 7, 8 9 in the grid as 'not yet colored' and solve those colors (knowing they represent a 7, 8 or 9). Then you will find that 'Blue' for example represents 9, red 8 and green 7 and no substitution is required in the end. Still, I love it.

This was amazing. The logic helped me to figure out a way around a roadblock in my career. Thanks Simon!

15:15 now that's a good use of the "i PeEl oFF tHe StiCkErS HahaAAaa" thing

@leojf3828

4 жыл бұрын

Me too. I LoL'D!

This.....this is simply brilliant. I have learned more in two videos from you about Sudoku than I have picked up in YEARS of playing Sudoku. Awesome

I managed to solve this on my own. I noticed that there was a single 9 in the upper row possibilities, and that solved it. Thanks for all the tips and tutorials

@FazalFariz

4 жыл бұрын

Could you please explain it? I'm still struggling to find a 'realistic' method of solution.

@freddykabuffke461

4 жыл бұрын

@@FazalFariz I just filled in what I could and came to a point, where i had three spots in the upper row wit 7/8, 7/9 and 7/8 Since two of the fields had to be a 7 or 8, the third one (or better, the second one) could only be a 9 Same with the right side i - 1 to i - 9, if you want. At one point theres only one square, where the three could go

@freddykabuffke461

4 жыл бұрын

@@FazalFariz So I tried again, and after one and a half hour, I'm still stuck at the remaining threes and fours as well as the sevens, eights and nines. Xwing and skyscrapers don't help here. So, I probably took a 7, 8 or 9 back then and pulled a "what if". Only thing I remember is, once I found out where it had to go, the rest was easy.

So how would one recognize when this is applicable? A puzzle where 2-4 digits are almost impossible to fill? Find a row, column or box where they exist and switch the values to empty boxes?

It’s easier to wrap your head around by using other standins such as the letters A, B, C, etc. then it becomes really obvious that you are dealing with temporary standing that should not be confused for a final solution.

Few days ago this channel popped up as recommended, no idea why. Also no idea why I watched sudoku videos all evening yesterday. Anyway I decided to give this puzzle a try before watching the video. I had attempted sudoku puzzles many years ago but always gave up or if I managed to solve them it was because I just had some lucky guesses. This time, using the logic and techniques I learned from watching the videos I was able to solve it in just under 22 minutes. Feeling proud.

I suppose maths minded people would find this technique more convincing if you just removed the numbers 7, 8 and 9 and put x, y and z in the bottom right box. Then at the end replace the x, y and z in the middle column with the corresponding original value and you’re done. Sort of solving for 3 variables using a sudoku. Pretty cool. It says a lot about the underlying maths of sudokus. You could possibly create an algebra of sudokus and have homeomorphic groups of solutions. I’m getting more and more into sudokus by the minute, cause of your channel :)

@Czeckie

4 жыл бұрын

mathematically minded people have no trouble understanding the symbol 9 as a symbol for the number 8 :)

@richardfarrer5616

4 жыл бұрын

You can do the same by replacing then with {purple, green, yellow} too. But yes, each puzzle is effectively a representative of 9! different puzzles for all the permutations of numbers.

@CoenBijpost

4 жыл бұрын

Czeckie yeah, agree. It does get a bit funnier if the 9 is a symbol for 8 AND 8 is a symbol for 9. Because in this case you’ve got an infinite loop in your representation. It’s just a bit of formalisation if you start working with letters to represent the replaced numbers, bit easier on the eyes as well, and no need for color representation during replacement.

@dolf370

4 жыл бұрын

@@Czeckie Sure, it just is easier to mess upp when you switch digits instead of having some other symbols which just makes it visually easier to keep track of what the heck you're doing.

This feels like a Kobayashi Maru workaround trick to me! :D

I don't know if what I'm going to say is completely correct but this technique reminded me of differential equations. If you want to solve any differential equation there are an infinite number of solutions and to reduce to a unique solution it is needed to give a boundary/initial condition. In the puzzle, you have a number of possible solutions and can permutate the 7, 8 and 9. But when you fixed the 7, 8 and 9 in the central column you are giving an initial condition to your problem and finding the exact solution for it. Its amazing.

13.51 - first time I'm timing these and posting around. I specifically avoided using the modified link. Instead I colored black the 3 digits I relabled (you can thus no longer see their original values in the grid), and I fixed them whenever i had to place them in the original squares (fortunately this was towards the end). It was nice to have to do this before fully solving 3s and 4s, and I'm really curious if I'd ever spot this in the wild. It was probably easier than one would normally find it, knowing the technique in advance, and having expected a video with an example of this technique. Nicely done!

🤣🤣🤣🤣 This is complete BONKERS ... and ABSOLUTELY GENIOUS !!!! I am speechless ... gosh, wow ... and all that! 👍👍👍 Cheating? No. Using computers is cheating. This is just yet another technique! Thanx for your videos! I only solve the classical ones every now and then, and I always return to this channel for the latest and greatest. This one was hilarious!

Learned something absolutely new and unique today. Thanks he says.... still scratching his head...

Very cool! How did you determine which box was the most promising? That's where I got stuck: I kept trying different ones but couldn't figure out which box would lead to the best results.

@nigelm5777

4 жыл бұрын

I think you can get the same result mapping to any row, column or box with only three unknowns. There are several possibilities, try one.

@CrimFerret

4 жыл бұрын

It doesn't matter. You don't have to wait until the end. If I'm not mistaken, at any point you should be able to remove all instances of the original numbers you removed wherever they appear in the puzzle and place them again in a different column row or box without affecting the solution for all the other numbers in the puzzle. You'd still be solving one of the same subset of puzzles. As long as once you solve it, you do the final swap back, you should have the solution.

@thereoncewasalimerickwriter

3 жыл бұрын

It's the only box with three cell empty and no 7, 8 or 9.

I'm here from the Greatest Hits Book. This is such a great educational video! You weren't kidding about this strat sticking with you. I'm gonna be remembering this for ages. Hopefully I can use it in future puzzles!

Can always just use A,B,C and you don't have to worry about people being picky. Just say A=8, B=9, C=7 after you solve it all.

@10:35, when Simon starts the new technique, wouldn't it be simpler to notice that whatever appears in R7C7 must also appear in R9C3. Using purple to mark those two cells, I was able to immediately identify all the other purple cells in the grid (in order - R6C2, R5C4, R4C8, R1C9, R3C6, R2C1) and see that they must all be 8.

Loved this demonstration of the repositioning method Simon. Thanks!!

Wow, a truly fascinating technique. Despite being a seemingly simple puzzle with rules not too difficult to understand, there really seems to be enormous depth to Sudoku strategies that I could not have figured out on my own, and this channel has been both really informative and enjoyable in that regard. Huge thanks for all the videos.

I was pondering the question of how many unique sudoku (basic) puzzles existed and the question arose was it a unique puzzle if two numbers were simply swapped. This technique seems an extension of the observation that the numbers are significant as positional identifiers rather than values.

That is really clever! (I might even be able to remember it.)

Very illuminating. Difficult to use on a pen and paper version.

@Kyrelel

4 жыл бұрын

Use letters instead of numbers, it makes the substitution easier at the end

Wow. That's an insanely great method. I don't know how often you can use it. But what a trick to have in the bag

I LOVE THIS!!!! Crazy logic!!!!!!!!!!! Beautiful.... Life will never be the same. It is an amazing concept . Thank you

That's quite a slight of hand. I'm just pleased I got to the same "stuck" point that you did before I started watching the video. Thank you and staff for these videos - they're great exercise. And, hello from the US.

I enjoiyed....I did! Wonder how many puzzles I could solve using that. Never seen the Ian's Plod techniques used. Get the puzzle resolved as far as you can.....look for a cell that has a binary choice that's crucial. Chose your best option...keep going till you finf it works or it doesn't, if it does, yippee...if it doesn't you know it's the other choice. Simples.

That one was astounding. I couldn't believe it. Thanks.

oh my gosh.. normally take an hour to do these puzzles.. strewth did this one in 14 minutes using this idea... am beaming :D

I'm not all that good at it, but I think a different more useful way to look at it, would be to give them non-numeric labels, like A, B and C, and solve with some arbitrary A, B and C placed. Then when you get to the point where you can prove that A, B or C would have to equal a specific number, you would actually replace them all. This might require some special support from the software to type it in, but that would be a less confusing way to do it. And that way could actually very easily generalize to situations where this way seemingly doesn't apply as is. And would be way easier to do on paper.

This is absolutely mind blowing. I mean, my brain can see that the ones you're relabeling in the middle row are... well, labels. you could label them A B and C for all it mattered. What I'm NOT getting is how you settle on the 789 in the middle as being re-labeled.

@chessw1

4 жыл бұрын

Isn't it because the 7 8 and 9 occur ONLY THERE, only in that one column? And isn't that what makes the technique rare, not often available for use? You have to get into a situation where certain numbers occur in a column, row, or box ONLY ONCE -- so you won't find many times when you can use the technique, but when it IS usable, it's very good.

It's incredible how this simple yet ingenious technique has never been discovered before...

@Kyrelel

4 жыл бұрын

It has. It is a common technique (in reverse) when constructing puzzles to rule out alternative solutions. i.e., if you have a puzzle with more than 1 solution, you keep adding digits until it's unique. Of course, people who knew/understood this also knew that you could do it backwards, i.e. removing digits, to get an ambiguous solution, you then refill with non-numbers (e.g ABC, keeping a note of which number is which letter) and then replace ... they (we, lol) just kept quiet about it :)

After this video, i am convinced that this algorithm really is applicable. Thanks for sharing.

This was a really cool puzzle. I was able solve it up to where there was only 7,8,9s left. In box 2 there was the two 7,8 pairs, so I did the 50/50 guess route. First guess I put the 8 in c4r2 and tried to solve, I got most of the way done and realized that was going to have 2 9s in box 9 so I hit the undo button back to my original guess changed it to 7 and was able to finished the puzzle. I really liked the solution in the video it would have made my guess work unnecessary. Thanks a bunch for your channel I have learned a ton of new strategies. :-)

@waltercreery4200

2 жыл бұрын

I also solved all of the 1-6's. I chose c1r2 for my 7/8 pair. I think this route was better than the solution in the video due to having a 50/50 of getting it right the first time versus a 33/33/33 chance.

I dont presume to be very good but after watching a few other technics you have shown on other videos I did this in 23minutes I didn't think it was that hard... so thank you for educating me on how to solve some harder Sudoku's

Basically it's algebra-like. You're solving for x, y, and z, but representing them by 7,8,9. I think the real trick would be recognizing when this is useful and which ones you can fill in to get an easier solution.

9:15 is as far as I got on my own. Couldnt figure out how to solve 3/4 or 7/8/9

22:30 for me on the initial version. Not exactly sure where the transformation was supposed to be realized, so just worked by placing the 3s and 4s on the end state in the most convenient positions, and then placed the 7,8, and 9 to see if those positions would interfere with my placing. They didn't.

I think this technique can also be helpful in situations where you don't have a unique case of three numbers, but stuck still with many of the same triples. And improved upon. Taking a suggestion from below, using colors for the temporary labeling instead of guessing which is which and later swapping. You can keep the 789 where they are, just work your way onwards with a yellow, green and blue until any of the colors resolve into a 7, 8 or 9. If it doesn't help wipe them and start again with another set of three squares... OR ... add another set of three colors! Add orange, red and purple and continue with these temporary labels until any resolves into EITHER 7, 8 or 9 OR into yellow, green or blue. Will this come in handy in many more situations?

The unit forcing chain looks similar to a technique you used on the New York Times one where you recognized three numbers in a row and penciled in a whole bunch after that.

Very clever! Congrats! Thinking outside! Very insightful! Brilliant does not even come close as a description!

I tried the initial version of the puzzle, and solved it. I bifurcated twice. Also, I did a loop (my term -- Y-wing perhaps?) eliminating one digit from a cell with three digits. I haven't yet listened to this new technique. I take it, the issue is resolving the 789s. Removing the 8 and 9 means that it's harder to place 5s and 3s. They were occupying potential a possible 5 and 3.

It took me 01:24:23 to solve that puzzle, admittedly I was getting distracted and doing other things at the same time. I'm not any good at spotting x-wing variants or chains at the moment. Once it got hard, I assigned colours to numbers and used them to try out possible solutions and delete notes when they didn't work in a solution. I'm pretty bad at the moment, but still pleased I even solved it at all haha.

Crazy that that works. Crazy.

I often find that handcrafted puzzles betray the creator's thoughts by assigning special roles to the low digits. This can easily be masked, and thus another level of difficulty added, by reshuffling the digits.

This one's interesting in that there's a free action on the 7s 8s and 9s in the sense of group actions. This in general will give you a hard problem no matter what. As long as you can get a free action around you'll make something like this.

This huge. Why isn't this on national television?

What would interesting to know is what percentage of all the line, column and box replacements (I counted 14) would result in an easy solution.

I paused halfway through when I knew what the secret was and I added ABC to the lower right-hand box and solved from there, and then when I figured out what a b and c were I just that way and it was probably easier and less confusing

OK, I tried solving it before watching your video. I found the colouring in your software very useful. I used "superposition" and solved all the 789 squares simultaneously without resolving which was which. After filling in all the blank squares with different colours and no collisions it then became obvious which numbers had to go where (including the 3's and 4's. Cute puzzle. Here is the "unique colouring" solution: imgur.com/a/O6syt5K

@theatog

4 жыл бұрын

This is essentially the same technique he was introducing. I think of it this way: Instead of having 7,8,9, you just replace them with XYZ. Then if you consider X, Y, Z can be any of 789, the XYZ is your "superposition" equivalent.

Reminds me of solving systems of equations in linear algebra. Very cool. Just relabel them x, y, and z then solve.

I found this technique less unnerving once I twigged I could use 'A', 'B' and 'C' to substitute the mystery digits (just as others have used colours). After that it's an 'algebra', and I found it much less scary. BUT let's not kid ourselves, this is a disguised bifurcation (or trifurcation) method, which is resolved at the end by swapping the numbers (or colours, or letters) back to fit their starting positions. So it's a bit naughty...

Very interesting technique. Though I solved it in about 22 minutes with chaining using the double notation ability in your app. A side note, you could have changed all those values in one click when you already had them highlighted, with clicking Normal then the number you wanted in there.

I too found this from the Greatest Hits book - and I used colouring on the 789 in column2 to find that one of those must be 9 and then went back and then coloured the 789 in box 9 differently, and this worked itself out. not something you can easily do on paper in the actual book...

I'm curious to see if someone will ever make a puzzle that uses this technique even more drastically. For instance, could there be a sudoku with an extreme solver rating that starts with exactly 25 clues: all the 1s and two of each other number? In this case, you could possibly solve the puzzle with this technique by erasing all of the givens that are not 1s and replacing those 16 numbers wherever you like in the grid. Then you are working with 8! = 40320 equivalent puzzles at once (and the "label cells" pick out which one is the true puzzle) instead of 3! = 6 as in this video. Probably this is impossible since the sudoku would very likely be underdetermined with these stipulations (there are sudoku's with a unique solution with only 17 givens, but these are far from having the hint multiplicities I'm describing: for instance, being given all nine 1s doesn't restrict the number of solutions as much as 9 generic hints tend to). In that case, what about a puzzle with all 1s given, all 2s given, and two of each other number given for a total of 9+9+2*7 = 32 givens? Then this technique would let you erase 14 givens and place them as you like, and you would be working with 7! = 5040 equivalent puzzles at once. Probably one more level is the point where I'm very confidant such puzzles with unique solutions exist, but I doubt they would be as difficult for the solver as I want them to be. In that case, it's 9+9+9+2*6 = 39 given, almost half the grid, and this technique makes you work with 720 equivalent puzzles at once. I would be happy to see a puzzle setter create any of these. Another thing I've wondered about is the possibility of making puzzles where doing this technique twice in parallel, or doing it recursively is advantageous. For instance, erase your given "label" digits and place them elsewhere in two different ways, working on both versions at the same time to glean information from both. Finding the value of some cell in both puzzles lets you translate what one number is represented by in the other. Alternatively, use this technique, and suppose you find all of the 7s for instance as a result. Then you can do the technique again on your original set of labels but without the 7 label included.

So what would be the criteria for a sudoku to justify, or even enable, the usage of this strategy? As a beginner, I was simply blown away by the logic of the strategy, it's simply ingenious. Though, I fail to really understand when it should be used, or if you could even, theoretically speaking, always use this strategy to solve a sudoku? Please excuse my noobiness, I'm just trying to understand, thanks!

If you remove the initial 8 and 9, then add 7, 8, 9 to box 9, the solver can't grade it since there are multiple solutions. I tried a few other things (like partially solving the puzzle, but it stayed on easy.

I was able to solve the original in reasonable time (17:16) by what I assume is an application of this technique in another guise. Basically, you can quickly get it to the point where you're drowning in unresolved {7,8,9}s. I picked a starting point, colored everything that was forced to be the same as that square, and picked colors for two types of squares that couldn't be the same as it. Pretty soon an 8 got resolved, and it snowballed from there.

that is just brilliant

let 7 be x ; let 8 be y ; let 9 be z .....easier to plug in and change at the end...I think the method you are showing is genius

@gunnarparment5050

4 жыл бұрын

Or A, B and C. Can also be done by using pen and paper or a generic software like a spreadsheet. (MS Excel, Google Calc, etc)

This feels like Algebra. You should do an example of this again, but use X Y and Z in place of 7 8 and 9. It will make it easier to replace them at the end too.

At about same situation as 9:33 I had a little bit more marked and noticed that there's only two possibilities on 8's at the bottom row. I selected the first eg. 8 in r9c3 and the whole puzzle collapsed. That might be called bifurgiation but I don't actually know how that differs from what relabeling is. I assumed that the 8 is false and I need then to relabel the 8's etc. but it actually was the correct one and I got lucky.

I like to just use colors to represent 7,8,9 … and eventually see which color goes with which number.

@hugo154

3 жыл бұрын

Thats how I solved it

Extremely useful.

hey there :) i've been looking for a practical web-interface for solving sudokus. the problem is just that i would like to insert my own sudokus and solve them (no need to check if they're solved correctly tho). your website seems perfect for that but it seems like i can only solve those sudokus from your videos there. is there access to a blank version or a similar app? i would be glad to hear from you :)

@Qazqi

4 жыл бұрын

I'd love this, but there's a possibility it could open them up to DMCA takedown requests or other legal hassles if people can upload copyrighted puzzles. If this is a concern, a short-term solution of not being able to generate a link to the puzzle would be fantastic for me. Coupling this with a simple import/export system (e.g., "__2__1_4__51..." for this puzzle) would be great and might be a way around DMCA.

@Aimlesswaves.

4 жыл бұрын

@@Qazqi Are sudoku's really copyrightable? Seems like just an almost arbitrary numberset, of such small value.

@Qazqi

4 жыл бұрын

@@Aimlesswaves. The channel has demonstrated how sudokus can be a creative work. I can only imagine they would fall under regular copyright laws the same way as other creative work. In my country, that includes protection for work you don't explicitly copyright. Anyway, it's just a guess; absolutely don't trust a random KZread comment on legal issues.

@jonpaxman

4 жыл бұрын

@@Qazqi it wouldn't need uploading, just a blank grid in which you can insert whatever sudoku you are working on.

@Qazqi

4 жыл бұрын

@@jonpaxman Sorry, I should've said hosting, as in being able to go to a URL with that puzzle like they do with the ones in the videos. In a way, you have to manually upload the puzzle by typing in the numbers, but I realize that's not a very conventional use of the word. If I can enter a puzzle and can't share a URL to it, I'm more than happy.

Oddly, I had no trouble solving this one using perfectly standard stuff picked up on this channel!! (Thanks Simon!). Nothing I'd consider crazy at all...I did have to do a couple short chains...that was it though.

The solution is basically to guess the placement of the three difficult numbers 7,8 and 9. And test the outcome. If the first guess is not the right one, try a different placing of 7,8 or 9. Easy peasy. No need for all that mumbo jumbo.

Actually, there is a simpler method❗ The initial version can be easily solved by coloring with three different colors the cells restricted to 7-8-9. After pencilmarking all possible *7-8-9 triples,* I started coloring them from box 9. I did not need to delete the given digits 7, 8, and 9. I did not need relabeling. And it was fun❗ I guess coloring is not easy to implement in a digital Sudoku solver, whereas for a human solver it is one of the simplest techniques. See also the comment by *Richard Farrer* below.

I solved it without with the original version of the puzzle. What I did was you highlights to connect square that I knew would be the same numbers even if I didn't know what they would end up being, allowing me to construct what I would imagine to me massive chains. Only once did I have to use "trial and error" to continue. Really all I did though was see was pick two of my highlight colors, and pretend the were the same number to see if an error would arise. When there was an error, the whole puzzle fell into place from there. Essentially, there is no reason to actually remove the numbers, all you need to do is sort of string the empty squares together that must be the same, until you get larger enough strings. After that, you can just try merging them together until one sticks.

Mindblowing stuff.

Thank you very much , this is really amazing !

So this shows that it works if there is a single row or column with numbers that don't appear anywhere else. I understood why it works for that particular case. You're basically creating a subset of puzzles with the other numbers in their same locations and choosing an easier to solve puzzle from that set and then swapping things back to the original puzzle. Can it be used under any other conditions that might come up more frequently or would that get too difficult to keep track of? I'm also wondering if you could do a partial solution of the new puzzle, swap back and then solve the rest of the way. It seems like that might work as well since any numbers placed other than the ones you initially swapped would be in the same positions in either solution.

Why not just assume a 7-8 placement in box 2 (forcing box 5 & 8 placements) and work to a potential contradiction or completion. Removes the relabeling hassle.

Mindblowing, as I said in a previous comment, but it seems to me that the technique has VERY limited application - not a problem, as it is an advanced technique - but also a limited application for this specific puzzle. There are three places you could have selected to put your 7, 8 & 9. Box 9 (which you chose), Box 7 (which because of the 3/4 pair only needs the top three numbers), or Row 1. But substituting into Box 7 or Row 1 don't give any obvious or simple solutions, so what is it that indicates Box 9 as the choice?

Messy if you try this technique on a piece of paper ! I think you could yield the same net effect just by guessing a cell. Not sure what the advantage is.

And what do you do if the location you relocated the 789s to prove to be just as hard? Do you just try again in a different location? Isn't that equivalent to bifurcation?

@jdyerjdyer

4 жыл бұрын

Possibly, but in placing the 7-8-9 in a new location, you can look at the puzzle to identify better placements over others. For example, where he placed them in the bottom right box, you could place them in several different configurations (6 to be exact) and some of those immediately give you a digit somewhere else, but possibly not all of them. So you would look for ways to quickly eliminate possibilities. It would be unlikely (but not impossible) for a puzzle with this setup to have all the possible placements lead to extreme difficulty. This is because even though those permutations are interchangeable, the positions in other locations are not. In other words, if you had a box with 5 open spots and you are swapping out only 3, you couldn't use this box as a placement because you could be placing the numbers where they can't go. Because of this, when you identify a spot that you can put them in arbitrarily (a box with only those digits left), then when placing them, they face that box of 5 digits and could eliminate some of the possible placements allowing you to progress, where as the original placement didn't face the right (or enough) open cells to help. Sorry if this is too long of an answer, but it is the easiest way I could explain it without going into math and permutation theory. Trust me, you don't want to go there unless you like wrapping your brain into knots. (That's a math pun. :) )

@jdyerjdyer

4 жыл бұрын

@Fester Blats I think his question is that just like in bifurcation you might not make any progress, i.e. you pick an equally (or harder) permutation that doesn't lead anywhere or at best only helps a little. In both cases you might have to back up, or try another permutation, but as you say, it is different in that any permutation you pick, once solved can be swapped out for the correct permutation.

@jdyerjdyer

4 жыл бұрын

One interesting thing to note in using this technique, and this might be of particular use in speed solving, is that any digits gained besides the ones you are permuting can be kept. So say you place the 7-8-9 and only gain placing a 3 and a 5, but become stuck again. You can permute the 7-8-9 into another location, but the 3 and the 5 are still validly placed, so you don't have to back them out. This is because the different permutations of the 7-8-9 don't affect the placement of the 3 and 5. Once identified, they are valid regardless of the permutation. So unlike bifurcation where if you go down the wrong rabbit hole you have to back everything out, using this technique, even if you pick a harder permutation, you might gain value. :)

@glennmelven3414

4 жыл бұрын

@@jdyerjdyer Yes, this is exactly what I had in mind.

Very cool! I only dabble in Sudoku but this fascinated me. Subbed for sure :)

My God. I feel like my face is melting off after opening the arc...