ABSOLUTELY GREAT! SUDOKU SWAMI IS DEFINITELY ABOVE AVERAGE, HE IS HIGH UP THERE and could have easily been a great Chemist of Physicist. I was an A student but I had to watch this particular video twice and CAREFULLY to get it. What is great about this technique is that since there are 33 patterns if you "light them up" the candidates that is (as Swami says) you almost immediately recognize one of those patterns. And as a former aficionado of the Rubik Cube I like to detect the patterns. BY FAR THE BEST AND MOST THOROUGH SUDOKU COURSE ON THE INTERNET! And God knows I have looked, from Harold Nolte (who I like a lot) to Sudoku Guy (who makes me laugh) to Cracking the Cryptic (interesting but not systematic, not logically organized and some bifurcation) none of them is a match to SUDOKU SWAMI. I AM GOING TO ORDER MY T-SHIRT AND MUG!
This was an absolutely glorious tutorial. I've been looking at page after page of people trying to explain Empty Rectangles, and this is the first page I've seen that made it make sense. Thanks!
In three different occasions I googled what an empty rectangle was and I could just not understand it from the written explanations. This is the fourth time and I decided to bite the bullet and watch a video. I have skimmed this one for less than 5 minutes and I got the gist of it finally! Will be coming back to finish this later.
Empty rectangles are the most frustrating to learn but the most glorious when you do. Took me a year and a lot of reading. I think the “the empty rectangle” is the perfect description. But you need to step back a bit and take in the picture of them put together as you ingenuously did at the 10 minute mark. Indeed there are empty rectangles all over the place and each configuration has 4 empty small boxes that line up in a box. And this is best seen stepping back as you showed. But the exciting part is just beginning. It reminds me of astronomy and trying to find the North Star. You don’t find it by finding the small dipper but by finding the BIG dipper, which looks like a big empty box/rectangle in the sky. And then it points you to the North Star. Which is important when trying to solve the orientation of the stars at night. In a similar comparison, the Big Dipper and the small dipper are a significant distance apart. The North Star is an average star. And it is on the small dipper, an average small constellation. So the key is really the BIG dipper. Just like the empty rectangle box is the key to finding the one candidate/number to eliminate, which is found in a whole different “constellation”: the conjugate pair in 2 totally different boxes of the puzzle. Significantly and commonly the elimination of the candidate solves/unravels the most difficult of sudoku puzzles. And finding the North Star is the key to establishing orientation in the sky for those living in the northern hemisphere at night. Finding it is like the Holy Grail and inspirational, just like astronomy. But maybe I over-speak.
Great stuff. Your thoroughness really allows the viewer to cement a concept. Long? Yes, but definitely thorough! Thanks for all of your hard work making these videos.
Thanks for this! I could never even hope to spot these before watching your video but now I am confident enough that I may be able to spot these patterns.
I'm grateful for this video because I need all of the education I can get. I do hard puzzles of varying difficulties, but with intermediate knowledge. I've recently learned about empty rectangles and applied it to a puzzle, labeled as super fiendish, that I discovered contained one. I was thrilled when I found it, as the lead in wasn't marked, and able to use it to limit a digit to two candidates in a box. I then discovered another empty rectangle and was able to use it to identify a hidden pair by being able to eliminate a digit from an otherwise seemingly possible candidate cell. It's all thanks to you, Sudoku Swami, that I was able to tackle this puzzle that would otherwise have been above my head and finish the solve! Thank you!
Well it all made sense so off to an Annoying level puzzle and there were three empty rectangles and those eliminations unlocked the puzzle. Fantastic, I need much more practice but your brilliant coaching is really helping me on the way to mastering Sudoku. Each new technique I learn I want to spot and then I remember all the other techniques I now have in my arsenal! Great mental workout.
For Beautiful Custom T-Shirts & Coffee Mugs featuring the Swami Logo, and also for Selected Classical Piano Pieces played by me, now available via Digital Download, please visit the Sudoku Swami Gift Shop! sudoku-swami.shopify.com
I had seen empty rectangles referenced in a lot of solving videos but didn’t understand them. This was super thorough and helped me crack open a puzzle I was stuck on for a while. Great explanation!
This is the video I have been searching for on Empty rectangles!! Thank you so much. I even now get the concept of strong and weak links, which I didn't seem to get beforehand. Top marks.
Thanks. A concept properly isolated and explained - unlike most sudoku videos, where they just complete a puzzle and think they've covered it in there.
What I found helpful is to work the "problem backward" i.e. instead of looking for an empty rectangle first, I try to spot the conjugate pairs first (which is relatively easy )and then I try to find an empty rectangle and the appropriate orientation (thus I don't have to worry about Ls, Crosses, Periscopes, Mushrooms etc.) to help me make the elimination. This lesson is arguably one of my favorites. It reminds me of the Rubik cube to some extent. When fully internalized, one can improve in Sudoku by leaps and bounds. SUDOKU SWAMI SIMPLY THE BEST!
Great. Whatever works. You can look for the ER Pattern first or the Conjugate Pair first. It makes no difference. Whatever is easiest. I'm sure over time, you will begin to see them both ways, with no problems. Thanks for the kind words. Good luck!
Empty rectangles have become one of the steps I use to solve puzzles. The mistake I've been making is not rechecking as the puzzle progresses. Eliminating one candidate changes everything!!! Richard
I was surprised but the video length initially but that is worth it ! Great explanations , I was starting to see them at the end (spotting the pairs first)
Hi Mr. Swami. The real example at 42:44 is for candidate 2 but - unless I missed something (as I probably do) - it seems to also work for candidate 4 which forms an empty 'L" rectangle in the same block as candidate 2 and it has both conjugate pairs in the same cells as well... it seems that I could eliminate candidates A as 4 and solve for candidate B as 4 at cell R6C5? How is that possible without violating the rule of unique solution?
In Block 1, the ER on 2, is a Dual ER, which places a 2 in R5C6. The ER on Candidate 4, in that same Block, is only a NORMAL (i.e., single) ER, which by using the Conjugate Pair of 4's in Column 5, will eliminate the 4 in R6C3 ONLY. I believe you are neglecting the Candidate 4 in R6C9. This means there is no Conjugate Pair on 4 in Row 6.
I like these. It's a new technique for me and I find them relatively easy to find. I just used it on a puzzle I was stuck on. (I'm still stuck but it did give me an elimination lol)
Hi Swamiji: Your explanation of the reasoning behind Double ERs is called reductio ad absurdum (RAA) in logic and math: if a premise leads, via a valid chain of reasoning, to an impossibility, or an absurdity, if you will, the premise must be false! :-) Once again, you've explained ERs brilliantly -- and simply! Congratulations!
Cool. Yes, I quite often make that argument, but was not aware of the formal name. Thanks for telling me. I'll mention it in my next Video, where this concept will once again come into play. Thanks for your kind comment, and for your continued support.
@@SudokuSwami And I must confess I never used ERs as a solving technique! X-chains, XY-wings, XYZ-wings, URs and so on, yes, but ERs never!!! Now that you have said it, I've realised, Aha!, they must of course be occurring quite frequently, and cracking ERs should be part of my repertoire!!! Once again, thank you. Greetings from India!!!
Yes, ERs are quite common. But sometimes, if there are only two Candidates on a diagonal that makeup the ER Pattern, it can also be perceived as a Turbot Fish, which you might be inclined to see first. But it doesn't matter. They will both lead to the same conclusion.
@@SudokuSwami Turbot fishes I know well, though I tend to think of them only as 3-link AICs, just like I think of 2-string kites as just 3-link AICs, not fussing or obsessing over the names. I only think of Skyscrapers as an entity in themselves. The other two, just as 3-link AICs! This ER video must count among the best 'explainers' of a difficult topic!
I had a hard time with empty rectangles going in. The 33 shapes were confusing and I found myself wondering if this or that was an empty rectangle. Then I realized you had the basic definition (An empty rectangle occurs whenever a candidate is situated in only one row and only one column of a particular block). For me this is seminal. Now I see them much more easily. Thanks, John
A & B are a Conjugate Pair. ONE of them HAS to be True. If A is True, it wipes out the Row or Column of the Empty Rectangle leaving a set of Locked Candidates Type I that will eliminate Candidate C. If Candidate B is True, it eliminates Candidate C because it is in the same House. Therefore Candidate C is False in either case. There are really just THREE basic shapes: The T, the Right Angle, and the Cross. But if you consider all the rotations, then there are NINE shapes, because there are three Rows and three Columns in each Block. But there are 33 ways these things can appear and manifest themselves, when you include the variations that don't have all 5 candidates filling the particular Row and Column Cells. It is good to be able to recognize ALL 33. But I'm glad it finally made sense to you.
You are right. There are a multitude of empty rectangle shapes in just about all puzzles. However, the necessary conjugate pairs outside of the empty rectangles are not nearly as abundant.
Being new to this game I have learned that each of your tutorials may not give you the answer but gives you an answer. I would get frustrated after I watched a couple your tutorials before realizing that each was just a piece of a much larger puzzle. I'm not just kissing up but I think the logic of your course should be taught in high schools. Richard Vanderpool
Exactly. Each Tutorial is a piece of the "larger puzzle," i.e., the Big Picture. Well-put. Thanks for your endorsement, and if you ever have any questions, just leave a comment below the Video, or send me an email: sudokuswami@gmail.com Good luck!
On exersises i noticed that a lot of time same candidate can be eliminated by finned x-wing or finned swordfish using same conjugate pair. I just more used to them need to practice more with rectangles it looks like easier way
Funny how sudoky works, There is always couple or more different ways to do same elimination. I usually spot cojugate pair than look in that order x-wings - skyscrappers/ finned/sahimi x-wings - here i think for now will be empty rectangles - swordfishes/finned swordfishes
This is a newly revised version of the original Tutorial #17. As a result, all of the previous comments posted here have unfortunately been deleted. Sorry about that. But please feel free to write again! :-))
Wow, this logic is second to non. Absolutely loving all this mind bending logic, I can't wait to put it into practice. One thing, you may have already cone this Swami, but could you go through an expert puzzle, from scratch using your techniques? Stay safe and keep the sudoku coming.
At 30:06, the same conclusion can be drawn with a finned x wing of 8s in columns 1 and 6. However, the empty rectangle strategy looks like a cool approach to use with structures that come pretty close to finned x wings but have some blockage. I'm surprised I have never heard of this until now.
You are correct. This is quite often the case. Additionally, that same configuration can also be seen as a Turbot Fish, with the endpoints being R3C3 & R7C6, producing the exact same elimination of the 8 in R7C3.
For me is just and L or T, and that s is. I like to keep it simple. Two perpendicular directions. You are over explaining, but at least you do it right. :) I hope you other videos are short, because I can not deal with another like this. Usefull, but way too long. Thanks.
@@SudokuSwami I mean two perpendicular directions. That s all I have in mind. There are many techniques, I can not memorize if I don t keep it simple. Thank you for you work.
This was so helpful! If you do empty rectangles again though, would you do one where you stay on the same candidate within a given puzzle until all the empty rectangles have been solved for that candidate? When you say the candidate number I hit pause and try and find the rectangle and work it out before you give the answer. I keep finding (or I think I do, hahaha) at least one other rectangle on a couple of these examples but because you then move on after showing us one example, I'm left wondering whether the other solution I found was correct, or just me identifying non-existent empty rectangles. :) Worried the latter seems more likely and my confidence would go further if I knew whether I was finding them correctly when left to my own devices. In any case, this was a great overview and thank you for making this concept accessible! :)
In these examples, I am just demonstrating "RANDOM" instances of the technique under discussion. There may or may not be others occurring at the same time. Have faith in yourself. If you understand the basic principle, and follow my instructions, and obey the Rules, you cannot make a mistake.
Hi. I love this course. You have taught me a lot of strategies! I have a question about an example at 28:44. You find an empty rectangle in block 1 and use the conjugate pair in column 8 to eliminate R8C1. But what happens when you have another empty rectangle that overlaps it? For example I see an empty square in block 7 with a conjugate pair in column 8 which would eliminate R2C2. I can perform elimination R2C2->R8C1, but not R8C1->R2C2 because the empty rectangle in block 7 no longer exists after R8C1 elimination. So do we need to look for all possible empty rectangle eliminations first, then perform them in an optimal order? Do I understand the concept correctly? I know there are not real puzzles, and we are not meant to solve them, but just need to look at the presented pattern, but I think this example poses an interesting question.
Yes, you seem to understand the concept. This is just a random example. And you are correct that there TWO Empty Rectangles on Candidate 6 occurring at the same time at 28:44. So, if you see something like this, just do them one at a time. It makes no difference which order. The end result will be the same. If you solve the ER in Block 7 first, and eliminate the 6 in R2C2, the two remaining 6's in Block 1 become Locked Candidates, and they will negate the 6 in R8C1. Good luck.
EDIT: There is a mistake here as I am counting some possibilities twice. I will update once I figure out the solution. I calculate the total number of empty rectangle configurations for a given candidate in a particular block to be 432, taking into account all possible orientations, patterns, and number of candidates appearing. Proof: An empty rectangle contains a candidate in exactly one row, so there are 3 possible rows. There are 3 cells in the row, so we are up to 3×3 configurations. Similarly, there are 3 possible columns. There are only 2 cells (instead of 3) that the candidate can occur in that column because one of the cells is in the already chosen row. Now we are up to 3×3×3×2 configurations. The row containing the candidate has 2 cells which can either have or not have the candidate which gives 2×2 possibilities. The column containing the candidate has only 1 more cell that can either have or not have the candidate which is 2 possibilities. Therefore, the total number of configurations is 3×3×3×2×2×2×2=432. ⚰
Yes, your first calculation was incorrect. It was too high. This is a very interesting problem. You cannot simply do the probability multiplication, as you have apparently discovered, because 1.) you must eliminate all duplicates, and 2.) your method produces some configurations that do not qualify as Empty Rectangle Patterns. For instance, let's consider Block 2. If you had Candidate X in R1C4, that would be part of your 3x3 for the Rows, right? And then, if you had Candidate X in R3C4, this would be part of your 3x2 for the Columns, right? But this configuration would not qualify as an Empty Rectangle Pattern. Your solution produces many similar results, which do not qualify as ER Patterns. Here is my best analysis of this: There are SEVEN possibilities for any Row in the Block. For any Candidate X, it can appear in any combination of the three Cells. Let’s call the Cells A, B & C. So, Candidate X can appear in A only, B only, C only, both A & B, both A & C, both B & C, or in A, B & C. For Row 1, if you consider all the possible Column placements, you get 54 unique configurations. Then for Row 2, after eliminating duplicates, you get an additional 48 unique configurations. And then for Row 3, after eliminating even more duplicates, you get an additional 42 unique configurations, for a total of 144 possible unique configurations.
To answer the question you posed about empty rectangle configurations at 11:40, there are 162 placements for the candidates in empty rectangles. Proof: We are given that there are 9 configurations for the 5 blocks in question. In each of these blocks there either is, or isn’t, a candidate. We have to consider the row and the column separately (because there has to be at least one candidate in each). We have two sets of cells: one of three cells (set A) and one two cells (set B). (This works equally well for a cross. The only difference is that set B in a cross is discontinuous). There are 2 raised to the third power possible configurations for set A. There are 2 raised to the second power possible configurations for set B. We have to subtract 1 from the number of possible configurations each for the sets where the candidate is not present in any cell, and we have to subtract 1 from the number of configurations for set A for the configuration where the only cell with a candidate is in the same row (or column) as set B. Therefore, ((2**3) - 1 -1) * ((2**2) - 1) = 18 possible candidate configurations for each block configuration. Given that there are 9 block configurations, that leads us to 9 * 18 = 162 distinct configurations.
C#. Are you a pianist by any chance? :-) Judging by the wording in your "proof" I am assuming you are the same person who wrote to me via email regarding this same issue. I wish I had calculated this before making this Video, but I am quite certain the answer is 144. Your solution has 18 duplicates; 6 for the second Row, and 12 more for the 3rd Row. Or else it contains configurations that do not qualify. Try drawing them out.....and you will see what I mean. :-))
C#, please check your email......I sent you a spreadsheet of the 144 configurations. Let's face it, this is really a moot point, as it is only a matter of trivia. Knowing this answer does not in any way help us solve puzzles. But it's fun to talk about it for a minute! :-))
Technically, there is another variation you didn't cover but it's obviously the same as a right angle; eg 7 7 7 [sp] 7 [sp] = space but come to think of it; what about 7 7 [sp] 7 This satisfies the "only one row only one columm, but does NOT leave a rectangle
Yes. What you describe, will function as a Right Angle. And yes, it does leave a Rectangle, if you exclude all the Cells in the Row and in the Column. At the end of my Tutorial on Remote Pairs (#29), I demonstrate the 144 possible patterns for Empty Rectangles. ruclips.net/video/N8ozmhdCcm8/видео.html
Dear Pneumatic, all you need to know are the 9 Main Shapes (the Right Angle with 4 rotations, the T with 4 rotations, and the Cross), or the 33 Sub-Shapes as described in the Tutorial. The 144 distinct and unique configurations, are covered by the 9 Main Shapes, and/or the 33 Sub-Shapes. :-))
@@SudokuSwami LOL. OK, I'll get right on it. Actually, though I can solve about 70% of evil sudokus with what I know, I don't have much functional grasp of anything X-chain or higher....and the 30% I can't solve are just flat out dead ends. But I'm going back to your early episodes to review the fundamentals because your presentations are so crisp and direct, and I appreciate it.
Is that a real butterfly behind him or has it been superimposed? It looks a beautiful place too. (It's amazing what one finds to be distracted by when in class).
At 24:00, when you say cell A is false, couldn't Row 1, Column 7 also be the candidate(9)? If so, that would make cell B false and cell C true? Or am I missing something?
The 9's in Row 1 are a Conjugate Pair. ONE of them has to be True. If the 9 in R1C5 is TRUE, then that would create a set of Locked Candidates in Block 8, making the 9 in R7C8 False. If the 9 in R1C5 is FALSE, then the 9 in R1C8 must be True, which would again, make the 9 in R7C8 False. So the 9 in R7C8 is False EITHER WAY. Which means it can safely be eliminated.
There are 3 basic patterns -- the "L" (right angle), the "T" and the "Cross". Each is fundamentally different from another as the "L" is formed by two edges of the block, the "T" uses only one edge and the "Cross" uses no edges. From any of these basic patterns, removing any single cell maintains the basic pattern; and there are 8 ways to remove two and maintain the pattern. Therefore, there are 1 + 5 + 8, or 14, different arrangements per pattern. So, we have: "L" : 14 * 4 rotations = 56 "T" 14 * 4 rotations = 56 "+" 14 * 1 rotation = 14 Total: 126 different empty rectangles If you include the nine different possible candidates, then 1134 !!
Sorry, I did not mean to downplay your calculations. I'm just saying that if you can visualize which Cells can be removed and still keep the integrity of the Pattern, you only need to know the Three Basic Patterns. Good luck.
You can also remove 3 Candidates from any full Right Angle, T, or Cross. And there are a few different ways to do it. All you need are Two Cells to make the minimum ER Pattern. So I think your calculation might be too low. :-))
@@SudokuSwami I did not include them as removing 3 creates a strong link between the remaining two, making at minimum an AIC. However, including them, there are two options -- one utilizing a corner of the block and one not utilizing a corner. With a corner cell of the block, there are 4 non-linear cells to be the other. There are 4 rotations to 3 of them and 2 rotations for opposite corners. 4 * 3 + 2 * 1 = 14. Adding 14 to my previous count sums to 140. Not using a corner requires two adjacent sides of the block, of which there are 4 rotations. Adding these 4 to the count works out to the total of 144.◙
@28:10 you eliminate 5 from (6,9) but couldn't candidate 6 also be eliminated from (6,9) with the exact same reasoning? The locked-pair of 5/6 in row 3 and that 6 also satisfies the Empty Rectangle requirement in box 4. Getting the naked single of 3 in (6,9) is quite valuable I think because it will force a 5/7 locked-pair in (1,5) and (2,5), in turn forcing 2 in (1,6).
Hello Doug. And welcome to my Channel. Yes, you are right, there is simultaneously an Empty Rectangle Pattern on Candidate 6, using those same Cells, which would eliminate the 6 from R6C9, and leave a Naked 3 in that Cell, as you noticed. Please understand, that these are just random, isolated examples, for demonstration purposes only. It is not my intention to fully SOLVE the puzzles in the examples. Good luck, and thanks for Subscribing! :-))
In your example at 28:55 is there not also an empty rectangle in block 9 that eliminates r2c9? (I think this would also subsequently eliminate r9c7 as you do by c9 being the only possibilities left for block 9.)
Yes, you are right. Using the ER that I pointed out, after you eliminate the 5 in R9C7, the remaining 5's in that Block become Pointing Locked Candidates, which will eliminate the 5 in R2C9. If you use the Empty Rectangle that YOU have pointed out, after eliminating the 5 in R2C9, you are left with Claiming Locked Candidates in Block 9, which will eliminate the 5 in R9C7. So you see? You get the exact same two eliminations, no matter which ER you use. But please understand, in these Tutorials, I am only pointing out random examples. My intent is NOT to solve these puzzles. The examples are meant for demonstration purposes only. But it is apparent that you understand this concept, which is great. That is the objective. :-)) Good luck!
I'm still not understanding how you know where the rectangle starts (which cell is A) and where the rectangle ends (which cell is C). I'm not understanding why A is A and why C is C. Why isn't C, A? Why isn't A, C? LOL. That wasn't explained. You go in all different directions so from my perspective it looks like you are just picking and choosing (23:32-31:39). This doesn't help me to figure out how to do this on my own.
@@SudokuSwami I appreciate the quick response. I would not have commented if I didn't watch the WHOLE video numerous times. I highlighted that section b/c that is where the "real" examples are shown. That is how we learn after all. During that portion of the video is the only area where I have questions so that is what my comment is focused on. For the examples you used I have furthur questions. You do not explain FULLY, you don't tell us why the cells you've designated as 'C' are the ones we eliminate from. I want to know why those cells are 'C' and why they aren't 'A'. The REAL examples, not the fake ones. That is all.
@@SudokuSwami Not sure why you assume you're the only one I watch on this vast website. Also, not sure why you're responding to comments just to say "watch the video". Just turn off the comments or not respond if you're not willing to give additional feedback to your content. Seems redundant.
Dear Yola. I am trying to be polite with you. Over 58,000 people have watched this Video, and you are the first person who has not understood my explanation of this technique. I put a lot of time and effort into creating these Videos, and they are FREE for you to watch. I feel that my explanation of this technique is quite clear. I don't know how to explain it any better than I already have in the Video. Again, I wish you good luck.
That's not what I said. The prerequisite is that for any Candidate x, all instances of that Candidate in a particular Block, must all be confined to One Row and One Column in that Block, regardless if there are 2, 3, 4 or 5 Candidate x's. With this in mind, I think if you watch the demonstrations, it should all make sense. Good luck. (It's One Row AND One Column......not One Row OR One Column.)
@@SudokuSwami I know what you're trying to say, only that I think there must be a better and more concise definition. Consider at 6:07 block 8. Here the candidate 3 can be in one of 3 rows OR one of 3 columns, what is clear is that in the final solution the candidate will occupy one cell, which belongs to one row and one column. Saying then that it can only be in one row or 1 column doesn't make sense, neither does changing the logic to an AND. Since then by this logic even 2 adjacent cells in the same block in the same row or column would technically meet the requirement. What I will do is work out the actual definition which I think is more let's just say clear and get back to you for a review. I'm sure that for you the definition makes sense, but that's because you're already an extreme guru, for the rest of us mere mortals we will need a more explicit definition. EDIT: The actual definition believe it or not is going to take a lot more thought from my end and a deeper understanding of the technique. Maybe something like this : When scanning a block from left to right top down, after after first row in block is scanned, in 2nd and 3rd rows of said block, the first cell containing a candidate directly below above candidate is the only column where subsequent candidates can be found, if no candidates above are found, this cell takes the first position.
You are making this more difficult than it is. I have given the Universally Accepted definition of this technique. I understand your point, but this is not Calculus or Quantum Mechanics. If you accept my definition, and correlate it with the various "shapes" I demonstrate in the Video, (essentially the T, the Right Angle, and the Cross), it should make perfect sense. If you can come up with a more precise definition, then by all means, bring it on. :-))
@@SudokuSwami I mean understand my point here. How can you expect people to understand a rule when directly in the first example the shape identified completely contradicts the rule? So here it is the actual definition : Once ALL candidates for a block are provided, the vectors for the candidates will form in one row AND any cell containing candidates NOT belonging to this row must exist in the same column AND must not be the only cell excluded from the initial row of candidates unless the initial row count is greater than 1. I think that is the most concise I can get it, that's the actual rule.
With all due respect, you are misinterpreting the original (simple) rule that I laid out. Yes, at 6:07 there is a Candidate 3 in Row 9. But that is irrelevant. ALL the 3's in Block 8 lie entirely within One Row and One Column, (Row 7 & Column 5), leaving an "Empty Rectangle" composed of the remaining four Cells. And that is all that matters for this technique to work. Row 7 & Column 5 (i.e., One Row and One Column), together, contain ALL the Candidate 3’s in Block 8. This statement cannot be made in regard to Row 8, Row 9, Column 4 or Column 6. They do not qualify, according to the rule. If you can find One Row and One Column within any particular Block, that together contain ALL the instances of Candidate x in that Block, then that’s it. That’s all you need. That is the simple requirement for identifying the starting pattern for this technique. You don’t need all the convoluted caveats and conditions that you spelled out. You are over-thinking it. Good luck.
The Candidates of the configuration lie in one Row and one Column within a Block. The other four Cells in the Block form a rectangle, that DO NOT contain the Candidates, and are therefore "empty." I know that's a little confusing and obtuse, but that's the true answer.
So, in the case of a T-Shape for example, it looks like there are 2 empty rectangles, 2 empty cells under 1 "arm" of the T, and 2 under the other "arm". Correct? (Just trying to understand the term Empty Rectangle, not the rule)
Right. You have to use your imagination a little bit to see it. The so-called "Empty Rectangle" is always made up of 4 Cells. You are correct about the T-Shape. With the Cross, the Empty Rectangle would be the 4 extreme corner Cells, and with a Right Angle, the 4 Cells would be bunched together in a little square. For instance, if the Right Angle pattern was located in Block 1, Row 1 and Column 1, the Empty Rectangle would be R2C2 & C3, and R3C2 & C3. But but forget about that. With this solving technique, you need to focus on the Cells where the Candidates ARE, and not where they are NOT. Good luck.
A and B must be a Conjugate Pair, and A must be in either the same Row or the Same Column as the ER. If B and C are ALSO a Conjugate Pair, then you have a Dual ER, and you can work it both ways. This is fully explained in the Video.
ABSOLUTELY GREAT! SUDOKU SWAMI IS DEFINITELY ABOVE AVERAGE, HE IS HIGH UP THERE and could have easily been a great Chemist of Physicist. I was an A student but I had to watch this particular video twice and CAREFULLY to get it. What is great about this technique is that since there are 33 patterns if you "light them up" the candidates that is (as Swami says) you almost immediately recognize one of those patterns. And as a former aficionado of the Rubik Cube I like to detect the patterns. BY FAR THE BEST AND MOST THOROUGH SUDOKU COURSE ON THE INTERNET! And God knows I have looked, from Harold Nolte (who I like a lot) to Sudoku Guy (who makes me laugh) to Cracking the Cryptic (interesting but not systematic, not logically organized and some bifurcation) none of them is a match to SUDOKU SWAMI. I AM GOING TO ORDER MY T-SHIRT AND MUG!
If you enjoyed this Video, please don't forget to click the SUBSCRIBE button, and the Thumbs Up Icon. It will really help me out. Thank you!
This was an absolutely glorious tutorial. I've been looking at page after page of people trying to explain Empty Rectangles, and this is the first page I've seen that made it make sense. Thanks!
Glad to hear it....!
In three different occasions I googled what an empty rectangle was and I could just not understand it from the written explanations. This is the fourth time and I decided to bite the bullet and watch a video. I have skimmed this one for less than 5 minutes and I got the gist of it finally! Will be coming back to finish this later.
Empty rectangles are the most frustrating to learn but the most glorious when you do. Took me a year and a lot of reading.
I think the “the empty rectangle” is the perfect description. But you need to step back a bit and take in the picture of them put together as you ingenuously did at the 10 minute mark. Indeed there are empty rectangles all over the place and each configuration has 4 empty small boxes that line up in a box. And this is best seen stepping back as you showed.
But the exciting part is just beginning.
It reminds me of astronomy and trying to find the North Star. You don’t find it by finding the small dipper but by finding the BIG dipper, which looks like a big empty box/rectangle in the sky. And then it points you to the North Star. Which is important when trying to solve the orientation of the stars at night.
In a similar comparison, the Big Dipper and the small dipper are a significant distance apart. The North Star is an average star. And it is on the small dipper, an average small constellation. So the key is really the BIG dipper. Just like the empty rectangle box is the key to finding the one candidate/number to eliminate, which is found in a whole different “constellation”: the conjugate pair in 2 totally different boxes of the puzzle.
Significantly and commonly the elimination of the candidate solves/unravels the most difficult of sudoku puzzles. And finding the North Star is the key to establishing orientation in the sky for those living in the northern hemisphere at night.
Finding it is like the Holy Grail and inspirational, just like astronomy.
But maybe I over-speak.
Great stuff. Your thoroughness really allows the viewer to cement a concept. Long? Yes, but definitely thorough! Thanks for all of your hard work making these videos.
Thanks for this! I could never even hope to spot these before watching your video but now I am confident enough that I may be able to spot these patterns.
This is one of the most interesting presentation you have given to us.
Logic is beautiful.
I'm grateful for this video because I need all of the education I can get. I do hard puzzles of varying difficulties, but with intermediate knowledge. I've recently learned about empty rectangles and applied it to a puzzle, labeled as super fiendish, that I discovered contained one. I was thrilled when I found it, as the lead in wasn't marked, and able to use it to limit a digit to two candidates in a box. I then discovered another empty rectangle and was able to use it to identify a hidden pair by being able to eliminate a digit from an otherwise seemingly possible candidate cell. It's all thanks to you, Sudoku Swami, that I was able to tackle this puzzle that would otherwise have been above my head and finish the solve! Thank you!
Great. Glad to hear it. Good luck!
Well it all made sense so off to an Annoying level puzzle and there were three empty rectangles and those eliminations unlocked the puzzle. Fantastic, I need much more practice but your brilliant coaching is really helping me on the way to mastering Sudoku. Each new technique I learn I want to spot and then I remember all the other techniques I now have in my arsenal! Great mental workout.
For Beautiful Custom T-Shirts & Coffee Mugs featuring the Swami Logo, and also for Selected Classical Piano Pieces played by me, now available via Digital Download, please visit the Sudoku Swami Gift Shop! sudoku-swami.shopify.com
I had seen empty rectangles referenced in a lot of solving videos but didn’t understand them. This was super thorough and helped me crack open a puzzle I was stuck on for a while. Great explanation!
The best explanation I found. Miles ahead of other vids on the subject.
This is the video I have been searching for on Empty rectangles!! Thank you so much. I even now get the concept of strong and weak links, which I didn't seem to get beforehand. Top marks.
This technique really helps, and is not hard to recognize, much thanks to your effort as you show all the rectangle patterns, that's a lot work.
Thanks. A concept properly isolated and explained - unlike most sudoku videos, where they just complete a puzzle and think they've covered it in there.
What I found helpful is to work the "problem backward" i.e. instead of looking for an empty rectangle first, I try to spot the conjugate pairs first (which is relatively easy )and then I try to find an empty rectangle and the appropriate orientation (thus I don't have to worry about Ls, Crosses, Periscopes, Mushrooms etc.) to help me make the elimination. This lesson is arguably one of my favorites. It reminds me of the Rubik cube to some extent. When fully internalized, one can improve in Sudoku by leaps and bounds. SUDOKU SWAMI SIMPLY THE BEST!
Great. Whatever works. You can look for the ER Pattern first or the Conjugate Pair first. It makes no difference. Whatever is easiest. I'm sure over time, you will begin to see them both ways, with no problems. Thanks for the kind words. Good luck!
great stuff, this is the first time I understand ER well
Thank you. Very thorough - I watched it all.
Empty rectangles have become one of the steps I use to solve puzzles. The mistake I've been making is not rechecking as the puzzle progresses. Eliminating one candidate changes everything!!!
Richard
Truer words were never spoken! Yes, I like ERs too, especially because you can find them in a large percentage of all Puzzles.
I was surprised but the video length initially but that is worth it ! Great explanations , I was starting to see them at the end (spotting the pairs first)
After watched twice I finally got the logic. Time to go into the puzzles and practise it. Tq
This is really great way to level up my skills. Thanks a millions.
This is the eight explanation I have seen or read- this
is 100% - rest were hopeless.
So you mean I am NOT hopeless?? LOL :-))
Wow, amazing, pure poetry....
Will look for them from now, espacially the Duals
Thank you very much for this great teaching.
Glad you enjoyed it.
Empty rectangles. Wow!. I did not understand them before this tutorial. They are powerful.
You can usually find one or two of these in almost every puzzle, if you look carefully enough.
What are you talking about, buddy? I don't have a favorite technique. I HATE Sudoku! (Ha ha.)
Hi Mr. Swami. The real example at 42:44 is for candidate 2 but - unless I missed something (as I probably do) - it seems to also work for candidate 4 which forms an empty 'L" rectangle in the same block as candidate 2 and it has both conjugate pairs in the same cells as well... it seems that I could eliminate candidates A as 4 and solve for candidate B as 4 at cell R6C5? How is that possible without violating the rule of unique solution?
In Block 1, the ER on 2, is a Dual ER, which places a 2 in R5C6. The ER on Candidate 4, in that same Block, is only a NORMAL (i.e., single) ER, which by using the Conjugate Pair of 4's in Column 5, will eliminate the 4 in R6C3 ONLY. I believe you are neglecting the Candidate 4 in R6C9. This means there is no Conjugate Pair on 4 in Row 6.
I like these. It's a new technique for me and I find them relatively easy to find. I just used it on a puzzle I was stuck on. (I'm still stuck but it did give me an elimination lol)
Hi Swamiji: Your explanation of the reasoning behind Double ERs is called reductio ad absurdum (RAA) in logic and math: if a premise leads, via a valid chain of reasoning, to an impossibility, or an absurdity, if you will, the premise must be false! :-)
Once again, you've explained ERs brilliantly -- and simply! Congratulations!
Cool. Yes, I quite often make that argument, but was not aware of the formal name. Thanks for telling me. I'll mention it in my next Video, where this concept will once again come into play. Thanks for your kind comment, and for your continued support.
@@SudokuSwami And I must confess I never used ERs as a solving technique! X-chains, XY-wings, XYZ-wings, URs and so on, yes, but ERs never!!! Now that you have said it, I've realised, Aha!, they must of course be occurring quite frequently, and cracking ERs should be part of my repertoire!!!
Once again, thank you. Greetings from India!!!
Yes, ERs are quite common. But sometimes, if there are only two Candidates on a diagonal that makeup the ER Pattern, it can also be perceived as a Turbot Fish, which you might be inclined to see first. But it doesn't matter. They will both lead to the same conclusion.
@@SudokuSwami Turbot fishes I know well, though I tend to think of them only as 3-link AICs, just like I think of 2-string kites as just 3-link AICs, not fussing or obsessing over the names. I only think of Skyscrapers as an entity in themselves. The other two, just as 3-link AICs!
This ER video must count among the best 'explainers' of a difficult topic!
Thanks. I really appreciate your appreciation. Ha-ha.
Please visit sudokuswami.com for an Outline of the Complete Course and news about upcoming Videos! Thank you.
SUPERIOR PEDAGOGICAL SKILLS. BUT I MUST ADMIT I WILL NEED TO WATCH THIS VIDEO AGAIN AND PERHAPS AGAIN:)!
I had a hard time with empty rectangles going in. The 33 shapes were confusing and I found myself wondering if this or that was an empty rectangle. Then I realized you had the basic definition (An empty rectangle occurs whenever a candidate is situated in only one row and only one column of a particular block). For me this is seminal. Now I see them much more easily. Thanks, John
A & B are a Conjugate Pair. ONE of them HAS to be True. If A is True, it wipes out the Row or Column of the Empty Rectangle leaving a set of Locked Candidates Type I that will eliminate Candidate C. If Candidate B is True, it eliminates Candidate C because it is in the same House. Therefore Candidate C is False in either case.
There are really just THREE basic shapes: The T, the Right Angle, and the Cross. But if you consider all the rotations, then there are NINE shapes, because there are three Rows and three Columns in each Block. But there are 33 ways these things can appear and manifest themselves, when you include the variations that don't have all 5 candidates filling the particular Row and Column Cells. It is good to be able to recognize ALL 33. But I'm glad it finally made sense to you.
Thanks for the additional explanation and comments
No problem, John. That's what I'm here for. :-))
You are right. There are a multitude of empty rectangle shapes in just about all puzzles. However, the necessary conjugate pairs outside of the empty rectangles are not nearly as abundant.
Being new to this game I have learned that each of your tutorials may not give you the answer but gives you an answer. I would get frustrated after I watched a couple your tutorials before realizing that each was just a piece of a much larger puzzle. I'm not just kissing up but I think the logic of your course should be taught in high schools.
Richard Vanderpool
Exactly. Each Tutorial is a piece of the "larger puzzle," i.e., the Big Picture. Well-put. Thanks for your endorsement, and if you ever have any questions, just leave a comment below the Video, or send me an email: sudokuswami@gmail.com Good luck!
You are the best.
Who else lol'd hard at the 'mother and kangaroo' analogy? 🤣🤣🤣
Sir if you wrote a sudoku guide book I would buy it, and it would make a great gift for puzzle lovers.
Thank you for the suggestion. I am thinking about it.......maybe someday. It's a huge undertaking.
Empty rectangle really likes conjugate pairs（multi-cells） in a block that is working as strong link.
Vert good
On exersises i noticed that a lot of time same candidate can be eliminated by finned x-wing or finned swordfish using same conjugate pair. I just more used to them need to practice more with rectangles it looks like easier way
Funny how sudoky works, There is always couple or more different ways to do same elimination. I usually spot cojugate pair than look in that order x-wings - skyscrappers/ finned/sahimi x-wings - here i think for now will be empty rectangles - swordfishes/finned swordfishes
This is a newly revised version of the original Tutorial #17. As a result, all of the previous comments posted here have unfortunately been deleted. Sorry about that. But please feel free to write again! :-))
what??! only to have you correct yourself again!?
never!!!!
Merry xmas
you do some solid explaining
you covered that as well
thourough
Best one so far... great, thanks
Thanks, Noam. Don't forget to watch the Random Tips Videos, as well, for additional insights.
Wow, this logic is second to non. Absolutely loving all this mind bending logic, I can't wait to put it into practice. One thing, you may have already cone this Swami, but could you go through an expert puzzle, from scratch using your techniques? Stay safe and keep the sudoku coming.
Yes, I will eventually do what you suggest. Thank you.
At 29.10 also in block 9 is an L which eliminate 5 in R2C9.
I pause every time and look for empty square myself.
At 30:06, the same conclusion can be drawn with a finned x wing of 8s in columns 1 and 6. However, the empty rectangle strategy looks like a cool approach to use with structures that come pretty close to finned x wings but have some blockage. I'm surprised I have never heard of this until now.
You are correct. This is quite often the case. Additionally, that same configuration can also be seen as a Turbot Fish, with the endpoints being R3C3 & R7C6, producing the exact same elimination of the 8 in R7C3.
For me is just and L or T, and that s is. I like to keep it simple. Two perpendicular directions.
You are over explaining, but at least you do it right. :)
I hope you other videos are short, because I can not deal with another like this.
Usefull, but way too long.
Thanks.
You are leaving out the Cross Shape.
@@SudokuSwami I mean two perpendicular directions.
That s all I have in mind.
There are many techniques, I can not memorize if I don t keep it simple.
Thank you for you work.
If you want to keep it simple, all those shapes boil down to three shapes: the L, the T, & the Cross.
Well said 🏆
This was so helpful! If you do empty rectangles again though, would you do one where you stay on the same candidate within a given puzzle until all the empty rectangles have been solved for that candidate? When you say the candidate number I hit pause and try and find the rectangle and work it out before you give the answer. I keep finding (or I think I do, hahaha) at least one other rectangle on a couple of these examples but because you then move on after showing us one example, I'm left wondering whether the other solution I found was correct, or just me identifying non-existent empty rectangles. :) Worried the latter seems more likely and my confidence would go further if I knew whether I was finding them correctly when left to my own devices. In any case, this was a great overview and thank you for making this concept accessible! :)
In these examples, I am just demonstrating "RANDOM" instances of the technique under discussion. There may or may not be others occurring at the same time. Have faith in yourself. If you understand the basic principle, and follow my instructions, and obey the Rules, you cannot make a mistake.
Super merci
Hi. I love this course. You have taught me a lot of strategies! I have a question about an example at 28:44. You find an empty rectangle in block 1 and use the conjugate pair in column 8 to eliminate R8C1. But what happens when you have another empty rectangle that overlaps it? For example I see an empty square in block 7 with a conjugate pair in column 8 which would eliminate R2C2. I can perform elimination R2C2->R8C1, but not R8C1->R2C2 because the empty rectangle in block 7 no longer exists after R8C1 elimination. So do we need to look for all possible empty rectangle eliminations first, then perform them in an optimal order? Do I understand the concept correctly? I know there are not real puzzles, and we are not meant to solve them, but just need to look at the presented pattern, but I think this example poses an interesting question.
Yes, you seem to understand the concept. This is just a random example. And you are correct that there TWO Empty Rectangles on Candidate 6 occurring at the same time at 28:44. So, if you see something like this, just do them one at a time. It makes no difference which order. The end result will be the same. If you solve the ER in Block 7 first, and eliminate the 6 in R2C2, the two remaining 6's in Block 1 become Locked Candidates, and they will negate the 6 in R8C1. Good luck.
I call the "thumbtack" pattern a "tetris"
you might be on too something there...
I will need to take my mid-term break here and digest this whole thing. Now I realize I was only doing elementary sudoku!
There is a lot more to Sudoku, than at first meets the eye.....
Spotting dual ER is most useful and convenient tools to eliminate cells!
It's fantastic, for sure; but extremely rare.
EDIT: There is a mistake here as I am counting some possibilities twice. I will update once I figure out the solution.
I calculate the total number of empty rectangle configurations for a given candidate in a particular block to be 432, taking into account all possible orientations, patterns, and number of candidates appearing.
Proof:
An empty rectangle contains a candidate in exactly one row, so there are 3 possible rows. There are 3 cells in the row, so we are up to 3×3 configurations. Similarly, there are 3 possible columns. There are only 2 cells (instead of 3) that the candidate can occur in that column because one of the cells is in the already chosen row. Now we are up to 3×3×3×2 configurations. The row containing the candidate has 2 cells which can either have or not have the candidate which gives 2×2 possibilities. The column containing the candidate has only 1 more cell that can either have or not have the candidate which is 2 possibilities. Therefore, the total number of configurations is 3×3×3×2×2×2×2=432.
⚰
Yes, your first calculation was incorrect. It was too high. This is a very interesting problem. You cannot simply do the probability multiplication, as you have apparently discovered, because 1.) you must eliminate all duplicates, and 2.) your method produces some configurations that do not qualify as Empty Rectangle Patterns.
For instance, let's consider Block 2. If you had Candidate X in R1C4, that would be part of your 3x3 for the Rows, right? And then, if you had Candidate X in R3C4, this would be part of your 3x2 for the Columns, right? But this configuration would not qualify as an Empty Rectangle Pattern. Your solution produces many similar results, which do not qualify as ER Patterns.
Here is my best analysis of this:
There are SEVEN possibilities for any Row in the Block. For any Candidate X, it can appear in any combination of the three Cells. Let’s call the Cells A, B & C. So, Candidate X can appear in A only, B only, C only, both A & B, both A & C, both B & C, or in A, B & C.
For Row 1, if you consider all the possible Column placements, you get 54 unique configurations. Then for Row 2, after eliminating duplicates, you get an additional 48 unique configurations. And then for Row 3, after eliminating even more duplicates, you get an additional 42 unique configurations, for a total of 144 possible unique configurations.
To answer the question you posed about empty rectangle configurations at 11:40, there are 162 placements for the candidates in empty rectangles.
Proof: We are given that there are 9 configurations for the 5 blocks in question. In each of these blocks there either is, or isn’t, a candidate. We have to consider the row and the column separately (because there has to be at least one candidate in each). We have two sets of cells: one of three cells (set A) and one two cells (set B). (This works equally well for a cross. The only difference is that set B in a cross is discontinuous). There are 2 raised to the third power possible configurations for set A. There are 2 raised to the second power possible configurations for set B. We have to subtract 1 from the number of possible configurations each for the sets where the candidate is not present in any cell, and we have to subtract 1 from the number of configurations for set A for the configuration where the only cell with a candidate is in the same row (or column) as set B.
Therefore, ((2**3) - 1 -1) * ((2**2) - 1) = 18 possible candidate configurations for each block configuration. Given that there are 9 block configurations, that leads us to 9 * 18 = 162 distinct configurations.
Unless I am misunderstanding the original problem posed, the answer is 162. See my reply to Swami's reply to Noel below for details.
C#. Are you a pianist by any chance? :-) Judging by the wording in your "proof" I am assuming you are the same person who wrote to me via email regarding this same issue. I wish I had calculated this before making this Video, but I am quite certain the answer is 144. Your solution has 18 duplicates; 6 for the second Row, and 12 more for the 3rd Row. Or else it contains configurations that do not qualify. Try drawing them out.....and you will see what I mean. :-))
C#, please check your email......I sent you a spreadsheet of the 144 configurations. Let's face it, this is really a moot point, as it is only a matter of trivia. Knowing this answer does not in any way help us solve puzzles. But it's fun to talk about it for a minute! :-))
Technically, there is another variation you didn't cover but it's obviously the same as a right angle; eg
7
7
7 [sp] 7 [sp] = space
but come to think of it; what about
7
7
[sp] 7
This satisfies the "only one row only one columm, but does NOT leave a rectangle
Yes. What you describe, will function as a Right Angle. And yes, it does leave a Rectangle, if you exclude all the Cells in the Row and in the Column. At the end of my Tutorial on Remote Pairs (#29), I demonstrate the 144 possible patterns for Empty Rectangles. ruclips.net/video/N8ozmhdCcm8/видео.html
Dear Pneumatic, all you need to know are the 9 Main Shapes (the Right Angle with 4 rotations, the T with 4 rotations, and the Cross), or the 33 Sub-Shapes as described in the Tutorial. The 144 distinct and unique configurations, are covered by the 9 Main Shapes, and/or the 33 Sub-Shapes. :-))
@@SudokuSwami LOL. OK, I'll get right on it. Actually, though I can solve about 70% of evil sudokus with what I know, I don't have much functional grasp of anything X-chain or higher....and the 30% I can't solve are just flat out dead ends. But I'm going back to your early episodes to review the fundamentals because your presentations are so crisp and direct, and I appreciate it.
Sounds great. Good luck! If you have any questions, please do not hesitate to ask. sudokuswami@gmail.com
Is that a real butterfly behind him or has it been superimposed? It looks a beautiful place too. (It's amazing what one finds to be distracted by when in class).
Everything is real.
thank you so much
At 24:00, when you say cell A is false, couldn't Row 1, Column 7 also be the candidate(9)? If so, that would make cell B false and cell C true? Or am I missing something?
The 9's in Row 1 are a Conjugate Pair. ONE of them has to be True. If the 9 in R1C5 is TRUE, then that would create a set of Locked Candidates in Block 8, making the 9 in R7C8 False. If the 9 in R1C5 is FALSE, then the 9 in R1C8 must be True, which would again, make the 9 in R7C8 False. So the 9 in R7C8 is False EITHER WAY. Which means it can safely be eliminated.
Wow! looking forwards to trying to spot these!
There are 3 basic patterns -- the "L" (right angle), the "T" and the "Cross". Each is fundamentally different from another as the "L" is formed by two edges of the block, the "T" uses only one edge and the "Cross" uses no edges. From any of these basic patterns, removing any single cell maintains the basic pattern; and there are 8 ways to remove two and maintain the pattern. Therefore, there are 1 + 5 + 8, or 14, different arrangements per pattern. So, we have:
"L" : 14 * 4 rotations = 56
"T" 14 * 4 rotations = 56
"+" 14 * 1 rotation = 14
Total: 126 different empty rectangles
If you include the nine different possible candidates, then 1134 !!
You are right: Right Angle, T, and Cross are all you need to know. Three Basic Patterns. The rest is just fluff.
Sorry, I did not mean to downplay your calculations. I'm just saying that if you can visualize which Cells can be removed and still keep the integrity of the Pattern, you only need to know the Three Basic Patterns. Good luck.
@@SudokuSwami No offense taken. I understand where you are coming from. May the best be with you.
You can also remove 3 Candidates from any full Right Angle, T, or Cross. And there are a few different ways to do it. All you need are Two Cells to make the minimum ER Pattern. So I think your calculation might be too low. :-))
@@SudokuSwami I did not include them as removing 3 creates a strong link between the remaining two, making at minimum an AIC. However, including them, there are two options -- one utilizing a corner of the block and one not utilizing a corner. With a corner cell of the block, there are 4 non-linear cells to be the other. There are 4 rotations to 3 of them and 2 rotations for opposite corners. 4 * 3 + 2 * 1 = 14. Adding 14 to my previous count sums to 140. Not using a corner requires two adjacent sides of the block, of which there are 4 rotations. Adding these 4 to the count works out to the total of 144.◙
@28:10 you eliminate 5 from (6,9) but couldn't candidate 6 also be eliminated from (6,9) with the exact same reasoning? The locked-pair of 5/6 in row 3 and that 6 also satisfies the Empty Rectangle requirement in box 4. Getting the naked single of 3 in (6,9) is quite valuable I think because it will force a 5/7 locked-pair in (1,5) and (2,5), in turn forcing 2 in (1,6).
Hello Doug. And welcome to my Channel. Yes, you are right, there is simultaneously an Empty Rectangle Pattern on Candidate 6, using those same Cells, which would eliminate the 6 from R6C9, and leave a Naked 3 in that Cell, as you noticed. Please understand, that these are just random, isolated examples, for demonstration purposes only. It is not my intention to fully SOLVE the puzzles in the examples. Good luck, and thanks for Subscribing! :-))
There is hope for me ... :)
In your example at 28:55 is there not also an empty rectangle in block 9 that eliminates r2c9? (I think this would also subsequently eliminate r9c7 as you do by c9 being the only possibilities left for block 9.)
Yes, you are right. Using the ER that I pointed out, after you eliminate the 5 in R9C7, the remaining 5's in that Block become Pointing Locked Candidates, which will eliminate the 5 in R2C9. If you use the Empty Rectangle that YOU have pointed out, after eliminating the 5 in R2C9, you are left with Claiming Locked Candidates in Block 9, which will eliminate the 5 in R9C7. So you see? You get the exact same two eliminations, no matter which ER you use. But please understand, in these Tutorials, I am only pointing out random examples. My intent is NOT to solve these puzzles. The examples are meant for demonstration purposes only. But it is apparent that you understand this concept, which is great. That is the objective. :-)) Good luck!
What is the name/composer of the piano piece at the beginning?
Etude Opus 10 #1 in C Major by F. Chopin.
What piano music is that at the beginning? That's really cool.
It is the Grand Etude in C Major by F. Chopin, Opus 10 #1
Great
I'm still not understanding how you know where the rectangle starts (which cell is A) and where the rectangle ends (which cell is C). I'm not understanding why A is A and why C is C. Why isn't C, A? Why isn't A, C? LOL. That wasn't explained. You go in all different directions so from my perspective it looks like you are just picking and choosing (23:32-31:39). This doesn't help me to figure out how to do this on my own.
You need to waatch the WHOLE Video from start to finish, and listen carefully to what I am saying. Everything is explained fully.
@@SudokuSwami I appreciate the quick response. I would not have commented if I didn't watch the WHOLE video numerous times. I highlighted that section b/c that is where the "real" examples are shown. That is how we learn after all. During that portion of the video is the only area where I have questions so that is what my comment is focused on. For the examples you used I have furthur questions. You do not explain FULLY, you don't tell us why the cells you've designated as 'C' are the ones we eliminate from. I want to know why those cells are 'C' and why they aren't 'A'. The REAL examples, not the fake ones. That is all.
If you are not able to understand my instructions and explanations, then I would suggest you find another teacher. Good luck!
@@SudokuSwami Not sure why you assume you're the only one I watch on this vast website. Also, not sure why you're responding to comments just to say "watch the video". Just turn off the comments or not respond if you're not willing to give additional feedback to your content. Seems redundant.
Dear Yola. I am trying to be polite with you. Over 58,000 people have watched this Video, and you are the first person who has not understood my explanation of this technique. I put a lot of time and effort into creating these Videos, and they are FREE for you to watch. I feel that my explanation of this technique is quite clear. I don't know how to explain it any better than I already have in the Video. Again, I wish you good luck.
The definition doesn't make sense that the candidate can be in either one row or 1 column.
That's not what I said. The prerequisite is that for any Candidate x, all instances of that Candidate in a particular Block, must all be confined to One Row and One Column in that Block, regardless if there are 2, 3, 4 or 5 Candidate x's. With this in mind, I think if you watch the demonstrations, it should all make sense. Good luck. (It's One Row AND One Column......not One Row OR One Column.)
@@SudokuSwami I know what you're trying to say, only that I think there must be a better and more concise definition. Consider at 6:07 block 8.
Here the candidate 3 can be in one of 3 rows OR one of 3 columns, what is clear is that in the final solution the candidate will occupy one cell, which belongs to one row and one column. Saying then that it can only be in one row or 1 column doesn't make sense, neither does changing the logic to an AND. Since then by this logic even 2 adjacent cells in the same block in the same row or column would technically meet the requirement.
What I will do is work out the actual definition which I think is more let's just say clear and get back to you for a review.
I'm sure that for you the definition makes sense, but that's because you're already an extreme guru, for the rest of us mere mortals we will need a more explicit definition.
EDIT: The actual definition believe it or not is going to take a lot more thought from my end and a deeper understanding of the technique.
Maybe something like this : When scanning a block from left to right top down, after after first row in block is scanned, in 2nd and 3rd rows of said block, the first cell containing a candidate directly below above candidate is the only column where subsequent candidates can be found, if no candidates above are found, this cell takes the first position.
You are making this more difficult than it is. I have given the Universally Accepted definition of this technique. I understand your point, but this is not Calculus or Quantum Mechanics. If you accept my definition, and correlate it with the various "shapes" I demonstrate in the Video, (essentially the T, the Right Angle, and the Cross), it should make perfect sense. If you can come up with a more precise definition, then by all means, bring it on. :-))
@@SudokuSwami
I mean understand my point here. How can you expect people to understand a rule when directly in the first example the shape identified completely contradicts the rule?
So here it is the actual definition : Once ALL candidates for a block are provided, the vectors for the candidates will form in one row AND any cell containing candidates NOT belonging to this row must exist in the same column AND must not be the only cell excluded from the initial row of candidates unless the initial row count is greater than 1.
I think that is the most concise I can get it, that's the actual rule.
With all due respect, you are misinterpreting the original (simple) rule that I laid out. Yes, at 6:07 there is a Candidate 3 in Row 9. But that is irrelevant. ALL the 3's in Block 8 lie entirely within One Row and One Column, (Row 7 & Column 5), leaving an "Empty Rectangle" composed of the remaining four Cells. And that is all that matters for this technique to work.
Row 7 & Column 5 (i.e., One Row and One Column), together, contain ALL the Candidate 3’s in Block 8. This statement cannot be made in regard to Row 8, Row 9, Column 4 or Column 6. They do not qualify, according to the rule.
If you can find One Row and One Column within any particular Block, that together contain ALL the instances of Candidate x in that Block, then that’s it. That’s all you need. That is the simple requirement for identifying the starting pattern for this technique. You don’t need all the convoluted caveats and conditions that you spelled out. You are over-thinking it. Good luck.
Why is it called Empty Rectangle? It's obvious when there's a square; otherwise, I don't see it. :(
Natalie
The Candidates of the configuration lie in one Row and one Column within a Block. The other four Cells in the Block form a rectangle, that DO NOT contain the Candidates, and are therefore "empty." I know that's a little confusing and obtuse, but that's the true answer.
@@SudokuSwami But
So, in the case of a T-Shape for example, it looks like there are 2 empty rectangles, 2 empty cells under 1 "arm" of the T, and 2 under the other "arm". Correct? (Just trying to understand the term Empty Rectangle, not the rule)
Right. You have to use your imagination a little bit to see it. The so-called "Empty Rectangle" is always made up of 4 Cells. You are correct about the T-Shape. With the Cross, the Empty Rectangle would be the 4 extreme corner Cells, and with a Right Angle, the 4 Cells would be bunched together in a little square. For instance, if the Right Angle pattern was located in Block 1, Row 1 and Column 1, the Empty Rectangle would be R2C2 & C3, and R3C2 & C3. But but forget about that. With this solving technique, you need to focus on the Cells where the Candidates ARE, and not where they are NOT. Good luck.
Thank you! So, the term Rectangle is a misnomer.
HOW DO I KNOW WHICH IS candidate A and candidate C???? A can be C and C can be A..how i need to decide???
A and B must be a Conjugate Pair, and A must be in either the same Row or the Same Column as the ER. If B and C are ALSO a Conjugate Pair, then you have a Dual ER, and you can work it both ways. This is fully explained in the Video.
It's sad that you have to do the disclaimer.
ER is a special type of turbot fish
this guy looks like michio kaku
Thumbtack=Tetris
Bingo!
Rats! I have seen several of the dual ER and did not it! Rats, rats! And, double rats.
Too much fluff
Super merci