Editor’s Note: When we started this series, I never thought Dan LeKander would give us 25 Puzzles, and if we don’t remind him, he may continue with 26…. and beyond. I certainly hope so. You may want to thank him… I am sure he will appreciate it!
As witnessed in the article last month we are on a roll with Step 6: of Dan’s Yes-No Challenge. This month will again be a celebration of “6”.
Puzzle # 25
First, we will revisit the approach as discussed in my first six TI Life articles.
DAN’S 8 STEP APPROACH TO SOLVING ALL SUDOKU PUZZLES
Once you have completed the puzzle, to the extent that you have filled in all obvious answers and have written all potential options across the top of the unsolved cells (PUZZLE PREPARATION), Dan recommends the following steps to complete the puzzle.
See TI Life Puzzle Preparation:
Step 1: Sudoku Pairs, Triplets and Quads – September, 2015
Step 2: Turbos & Interaction – October, 2015
Step 3: Sudoku Gordonian Rectangles and Polygons – November, 2015
Step 4: XY-Wings & XYZ Wings – December, 2015
Step 5: X-Wings – January, 2016
Step 6: DAN’S YES/NO CHALLENGE
Step 7: DAN’S CLOSE RELATIONSHIP CHALLENGE
Step 8: AN EXPANSION OF STEP 7
Steps 1-5 are relatively common techniques and are explained in the TI LIFE articles per above. Steps 6-8 are covered in detail, in Dan’s book.
Also see Sudoku Puzzle Challenge… February 2016, Sudoku Puzzle Challenge–March 2016, Sudoku Puzzle Challenge–April 2016, May 2016, June 2016, July 2016, August 2016, September 2016, October 2016, November 2016, December 2016, January 2017, February 2017, March 2017 and April 2017.
As a reminder, the basic rules of Sudoku are that the numbers 1-9 cannot be repeated in a row, column, or box, and there can only be one solution to the puzzle.
First, we will complete the 4 Steps of Puzzle Preparation …
1. Fill in the obvious answers
2. Fill in the not-so-obvious answers
3. Mark unsolved cells with options that cannot exist in those cells
4. Fill in the potential options for the unsolved cells
OBVIOUS ANSWERS … Start with the 1’s to see if there are any obvious 1-choice answers. Then navigate your way through the 2’s to 9’s.
The first obvious answer is C5R7=1 (cell in column 5, row 7). C3R6=1. C2R6=3. C6R8=3. C3R9=3. C1R9=2. C3R2=2. Now your grid should look like Example #25.1 below:
Example # 25.1
NOT SO OBVIOUS ANSWERS … In box 3 (grid of 9 cells in the upper right) a 9 can only exist in C7R1 or C7R3 (because of the 9’s in C2R2 & C9R9); therefore, a 9 cannot exist in C7R4, C7R5 or C7R6. In box 4 a 9 can only exist in C1R4 or C3R4; therefore a 9 cannot exist in C7R4 or C8R4. C9R9 being a 9 precludes C9R5 & C9R6 from being a 9. This leaves only 1 cell in box 6 that can be a 9. C8R6=9. Now your grid should look like Example #25.2 below:
Example # 25.2
NUMBERS IN UNSOLVED CELLS THAT CANNOT EXIST …
In box 5 a 7 can only exist in C4R4 or C5R4; therefore, a 7 cannot exist in C7R4 & C8R4. Indicate this by placing a small “7” in the bottom of these two cells, as this will be your reminder to later not include a 7 as an option.
In box 4 an 8 can only exist in C1R4 or C2R4; therefore, an 8 cannot exist in C7R4 & C8R4.
In box 8 an 8 can only exist in C4R9 or C5R9; therefore, an 8 cannot exist in C8R9.
Now your grid should look like Example #25.3 below:
Example # 25.3
FILL IN THE OPTIONS FOR THE UNSOLVED CELLS …
When we fill in the options for C8R9 we find that a 7 is the only option that can exist. C8R9=7. Then C9R2=7, C7R5=7, C4R8=7, C2R7=7 & C5R4=7.
Now your grid should look like Example #25.4 below:
Example # 25.4
You may note that we have an obvious pair with C4R9 & C5R9. It is always efficient to process an obvious pair as soon as spotted. In box 8 C6R7 cannot be a 5; therefore, C6R7=9. Then C1R8=9. C3R4=9. C4R3=9. C7R1=9.
Now note that in row 4 a 2 can only exist in C7R4. C7R4=2. The only cell in row 2 that can be a 1 is C1R2. C1R2=1. Now your grid should look like Example #25.5 below:
Example # 25.5
STEP 3. Gordonian theory: In Example #25.5 above the options of 4 cells are highlighted in yellow. This is a variation of a one-sided Gordonian Rectangle. The theory was explained in a previous article. If neither C1R5 or C1R7 is a 5, then those two cells would both have options 46. That would create a situation where there could be two answers to the puzzle which violates the rule of Sudoku that there can only be a single solution. Therefore, either C1R5 or C1R7 must be a 5. That being the case, no other cell in column 1 can be a 5, eliminating the 5 as an option for C1R4, giving us Example #25.6 below:
Example # 25.6
Step 2. Interaction & Turbos: Having eliminated the 5 in C1R4 above, we set up a potential Turbo in box 4. As marked in Example 25.6 above, C1R5 & C2R4 set up a Turbo with C9R5 & C2R8 to eliminate the 5 from C9R8. (Theory … if C2R4 is a 5, then C1R5 is not a 5, and then C9R5 is a 5, then C9R8 is not a 5. If C1R5 is a 5, then C2R4 is not a 5, then C2R8 is a 5, then C9R8 is not a 5. One of the two cells in box 4 must be a 5. In neither case can C9R8 be a 5, eliminating the 5 from C9R8). Now your grid should look like Example #25.7 below:
Example # 25.7
STEP 6: DAN’S YES-NO CHALLENGE
There are 3 circumstances that establish the potential for a Step 6 exercise:
1. Look for just 2 unsolved cells in a box that contain the same option and these 2 cells are not in the same row or column. An example is C4R4 & C6R6. Each of these 2 cells contain a 6 and are not in the same row or column.
2. Look for just 2 unsolved cells in a column that contain the same option and these 2 cells are not in the same box. An example is C4R2 & C4R9. Each of these 2 cells contain an 8 and are not in the same box.
3. Look for just 2 unsolved cells in a row that contain the same option and these 2 cells are not in the same box. An example is C2R1 & C5R1. Each of these 2 cells contain a 4 and are not in the same box.
Example # 25.8
We will start with box 6 and highlight C8R4 & C9R in yellow as the 2 “starting cells” per Example #25.8 above. These are the only two cells in box 6 with a 4 as an option, and the cells are not in the same column or row. We will assume first that C8R4 is a 4 and mark it as “Y”. It then follows that if C8R4 is a “Y” (a 4), then C1R4 & C2R4 are an “N” (not a 4). Next, we assume C9R5 is a 4 and mark it with a lower-case “y”. If C9R5 is a “y”, then C9R7 & C9R8 are an “n”, then C8R7 is a “y”, then C1R7 & C3R7 are an “n”, then C2R8 is a “y”, and then C2R4 is an “n”. We see that C2R4 is not a 4 (“N,n”) regardless of which starting cell is a 4; therefore, the 4 can be eliminated as an option from C2R4. Can we go further? Yes! If C2R8 is a “y”, then C2R1 & C2R3 are an “n”. Then C3R1 is a “y”. Then C1R4 is an “n”. So C1R4 is also a “N,n” and we can eliminate the 4 from C1R4 as per Example #25.9 below:
Example # 25.9
We now notice that the only cell in row 4 that has a 4 as an option is C8R4. C8R4=4.
From this point the puzzle is “unlocked” and is easily solved giving us Example #25.10 as the conclusion:
Example # 25.10
Recently a T.I. Life reader asked me how to best detect potential Step 6’s. Good question! First start with each box. When you find a potential Step 6, begin with 1 of the starting cells and use your fingers of one hand to designate the “no’s”. Then with the other starting cell go the other direction and use your fingers to designate the “no’s”. If a cell has a “no” from both starting cells, then it cannot be that number. Next finish searching all boxes. Then perform a similar search in all rows, and then all columns.
In very complex Step 6 exercises it will benefit to mark the yes’s and no’s in pencil, as was the case above. You will find that most potential Step 6 situations are “short-ended”, meaning you can determine almost instantly that they will not yield a clue. Some puzzles will yield no successful Step 6’s. Other puzzles may yield multiple successful Step 6’s. Once you learn how to detect all potential Step 6 candidate and follow the yes-no procedure it will become second nature to use and enjoy this technique.
When l discovered and began using Step 6 many times I would forget to check the boxes for potential Step 6 exercises. Shortly thereafter I realized that when you search for Turbos (Step 2) you are looking for the same two cells in a box as a Step 6 exercise. So, when I performed the Turbo search for potential candidates, I mark each of the 2 cells in the box with a small period in pencil in the bottom of the 2 cells. Then when I arrived at Step 6 all potentials are already marked.
Thank you for your interest. May the gentle winds of Sudoku be at your back.
(Photo above… Sudoku is for everyone… you’re never too young to start!)
Do you tackle a Sudoku on your cottage veranda, sailboat cockpit, or at a campsite? TI Life is taking full advantage of Dan LeKander, from Wellesley Island, who is a Sudoku expert and author of “3 Advanced Sudoku Techniques – That Will Change Your Game Forever!”
In January 2016, we published a final article in his original series – but many of us enjoy using “Dan’s Steps,” so when he asked if we would like a puzzle to solve every month … this editor said an enthusiastic… Yes, please!
I suggest you try this relatively difficult puzzle and that you also purchase Dan’s book, “3 Advanced Sudoku Techniques, That Will Change Your Game Forever!”
Most importantly, I ask that you leave comments on any part of his series and throughout the year. Remember when your teacher said – no such thing as a silly question – as we can all learn together.
Dan’s book is available online, amazon.com and on ebay.com.
Purchase of a book includes a 50-page blank grid pad, 33 black and two green tokens, to assist with Step 6.…
Now, with the new year well underway, I want to say THANKS Dan and THANKS to his better half, Peggy LeKander, who helps make sure the article is correct and reads well. What a team!
Dan LeKander and his wife, Peggy, have been seasonal residents of Fineview, on Wellesley Island, NY, since 1983. In addition to being a Sudoku addict, Dan explores the 1000 Islands to enjoy the wildlife, beauty and of course, Catch-and Release bass fishing.
[See Jessy Kahn’s Book Review, “3 Advanced Sudoku Techniques…” by Dan LeKander, June issue, 2015, of TI Life.]
Do you have suggestions for future articles or do you want any Step 1-8 techniques explained further? You can post on this article or contact me directly at firstname.lastname@example.org