SUDOKU IS FOR EVERYONE! FEBRUARY 2017
PUZZLE 22 is regarded as a very difficult puzzle. Again, a step-by-step guide accompanies this puzzle. So even if you are a novice, you can learn how to solve difficult puzzles, using Dan's 8 -Step Approach!
First, we will revisit the approach as discussed in Dan’s 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 and January 2017.
Puzzle # 22
Puzzle Preparation is quite involved with this puzzle. If you think you can solve it, I suggest you get your grid to the end of Puzzle Preparation (solve the unsolved cells that can be solved without using techniques 1-8, and fill in the options of the remaining unsolved cells). Then compare to Example 22.3 below. If you have the same grid, perfect. If not, you may want to discover what you missed, and why!
As a reminder, the basic rules of Sudoku are that numbers 1-9 cannot be repeated in a row, column, or box, and there can only be one solution to a puzzle.
First, we will complete the 4 steps of Puzzle Preparation …
- Fill in obvious answers
- Fill in the not-so-obvious answers
- Mark the unsolved cells with options that cannot exist in those cells
- 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 (a 1-choice answer means there is only one number of numbers 1-9 that will work in that cell). Then move on to the 2’s through 9’s.
The first obvious answer is C8R4=1 (cell in column 8, row 4), then C7R4=2, C3R9=2, C4R7=3, C7R5=3, C5R8=4, and C6R7=2.
NOT SO OBVIOUS ANSWERS … The only cell in column 1 that can have a 3 is C1R4.
In box 5 (middle box of 9 cells) the 9 in C6R8 precludes C6R4 & C6R5 from being a 9. The 9 in C8R6 precludes C4R6 from being a 9; therefore, a 9 must exist in C4R5 or C5R5. This precludes C1R5 or C3R5 from being a 9; therefore, the only cell in box 4 that can be a 9 is C3R4.
The only cell in column 7 that can be a 4 is C7R6; then C3R5=4.
The 8 in C2R4 precludes an 8 in C6R4. The 8 in C4R2 precludes C4R5 & C4R6 from being an 8; therefore, there is an 8 in C5R5 or C6R5. As a result, the only cell in box 6 that can be an 8 is C9R6.
Now your grid should look like Example 22.1 below:
Example # 22.1
NUMBERS IN UNSOLVED CELLS THAT CANNOT EXIST …
The 5 in C3R3 & C5R7 means that a 5 must exist in C4R1 or C6R1; therefore, C7R1 & C8R1 cannot be a 5.
The 5 in C3R3 & C5R7 means that a 5 must exist in C2R8 or C2R9; therefore, C2R6 cannot be a 5.
Now your grid should look like Example 22.2 below:
Example # 22.2
FILL IN THE OPTIONS FOR THE UNSOLVED CELLS …
In row 4, the options for C6R4 & C9R4 are 57.
In box 8 the only options in row 9 are 1,7 & 8, which is a triplet; therefore, a 1, 7 & 8 cannot exist in the other cells in row 9.
C9R4 & C9R5 both have options 57, which is an obvious pair; therefore, a 5 and 7 cannot exist in any other cells in column 9.
C1R5 & C9R5 both have options 57, which again is an obvious pair; therefore, a 5 and 7 cannot exist in any other cells in row 5.
Now there is a triplet of 189 in C4R5, C5R5 & C6R5; therefore, the 1 cannot exist in C4R6, making its options 57.
Now finish filling in the options for the unsolved cells and you have Example 22.3 below. Note that once again, much progress was made in the Puzzle Preparation phase!
Example #22.3 … through Puzzle Preparation
STEPS 1-8 … I suggest exploring Steps 1-8 in the order of 1 through 8. Please also keep in mind that when you discover a clue, it may then require you to look at previous steps for other clues, which happens 3 times below.
- PAIRS, TRIPLETS & QUADS (Step 1). As you examine the grid, you find no additional Step 1 clues.
- INTERACTIONS & TURBOS (Step 2). In Example 22.3 above, C1R7 & C3R8 (highlighted in yellow) form a Turbo with C3R1 & C8R7 (highlighted in green). Where the green cells intersect cannot have an 8 as an option; therefore C8R1 is not an 8 and can be eliminated as an option from that cell. (Theory refresher … you detect a potential Turbo by finding only 2 unsolved cells in a box that share a common option, but are not in the same row or column. Then find only 1 other cell in the row from one of the cells & only 1 other cell in the column from the other cell. Where they intersect precludes that cell from having that option. If C1R7=8 , then C3R8 is not an 8, then C3R1=8 and then C8R1 is not an 8. If C3R8=8, then C1R7 is not an 8, then C8R7=8, then C8R1 is not an 8. So regardless of which yellow cell is an 8, C8R1 is not an 8.)
- At this point, there are no additional Step 1-5 clues, and you now have Example 22.4 below:
Puzzle # 22.4
STEP 6 … DAN’S YES-NO CHALLENGE. Please note above there are only two unsolved cells in column 3, which contain an 8 as an option, as highlighted in yellow. This sets the stage for a potential Step 6. We start with the premise that one of the two yellow highlighted cells must be an 8. If C3R8 is 8, then C8R8 is not an 8. If C3R1 is the 8, then C7R1 is not an 8, then C8R3 must be an 8, (the only other choice for an 8 in box 3), then C8R8 is not an 8. So regardless of which cell in column 3 is an 8, C8R8 cannot be an 8, eliminating the 8 as an option for that cell. This situation is also called a “Finned X-wing”, which is usually difficult to detect without Step 6. To see this in the Yes-No format, see example 22.5 below:
Example # 22.5
The theory is the same as described above, however, with the Yes-No concept, you assign a “Y” to one of the starter cells and a “y” to the other. So if C3R1 is yes “Y”, C7R1 is no “N”, C8R3 is yes “Y”, and C8R8 is no “N”. If C3R8 is yes “y”, then C8R8 is no “n”. Any cell that ends up with a N,n cannot be that digit, an 8 in this example. The Yes-No approach is essentially imperative with more complex Step 6 situations.
Exampole # 22.6
Now your grid should look like Example 22.6 above, with the 8 missing from C8R8. Please notice there are only two unsolved cells in box 9 that have an 8 as an option, as highlighted in yellow. Those two cells, along with the two green highlighted cells, form a Turbo, eliminating the 8 from the C1R1. Now your grid should look like Example 22.7 below:
Example # 22.7
Next, notice the removal of the 8 from C8R8, two steps ago, has created a pair (57) in column 8, with C8R2 & C8R8; therefore a 5 and 7 can be eliminated in all the other unsolved cells in column 8. As a result of eliminating the 7 in C8R1, C8R1=2. Your grid should now look like Example 22.8 below:
Example # 22.8
Here we notice that we have created an Interaction, with cells C7R8 & C8R8. Since one of these two cells in box 9 must be a 7, you can eliminate the 7 in C2R8 & C3R8. Now your grid should look like Example 22.9 below:
Example # 22.9
On to Step 7 … Dan’s Close Relationship Challenge
To begin this step, we pick any unsolved cell with two options (2-option cell) and call it the “driver cell” as it will be the center of our focus. We will pick C2R9 as the driver cell, with the sequence 9,5 and mark the cell as such (highlighted in yellow in Example 22.10 below). At this point, we do not know if C2R9 is a 5 or a 9. We do know that if C2R9 is a 9, then C2R2, C1R7 & C2R7 are not a 9, and we will mark them with “N9” indicating so. Next we will assume C2R9 is a 5. If C2R9=5 we then know that at least one of the three cells, C2R2, C1R7 & C2R7 is not a 9. Can we determine this? Yes! We then “track” the 5 through the puzzle to see which of those three cells are not a 9. We begin the tracking with the 5 in C2R9 (highlighted in green), utilizing the 3rd level of the unsolved cells.
Example # 22.10
Please look at Example 22.11 below. You can see that if C2R9=5, then C7R9=6, C9R9=9, C9R7=4, C8R7=8, C8R3=4, and C9R3=6. Now, the options for C7R2 & C8R2 are 57; therefore, C5R2=9, C3R2=3, and C2R2=6. We will stop here to analyze what we have learned.
Example # 22.11
Logic: C2R9 must be a 9 or 5. If C2R9=9, C2R2 is not a 9. If C2R9=5, C5R2=6 and C2R2 is not a 9. Therefore the 9 can be eliminated from C2R2.
Continue the tracking. The options for C2R3 & C2R6 are now 17, which is a pair. Therefore, C2R7 cannot be a 7 and is a 9.
Logic: C2R9 must be a 9 or 5. If C2R9=9, C1R7 is not a 9. If C2R9=5, C2R7=9 and C1R7 is not a 9. Therefore the 9 can be eliminated from C1R7.
Example # 22.12
In further tracking, we see that C7R2, C8R2, C7R8 & C8R8 each have options 57. This is a Gordonian Rectangle, which creates a conflict! There could be 2 answers to the puzzle. Therefore, C2R9 is not a 5. We make C2R9=9, and you have Example 22.13 below:
Example # 22.13
By establishing C2R9=9, you have unlocked this puzzle! C2R7=7, C1R7=8, C3R8=3, C2R8=5, and the puzzle is easily solved from here.
Thank you for your interest. May the gentle winds of Sudoku be at your back!
(Photo above… Sudoku is for everyone… and 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 as we start a whole new year, 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 of TI Life.]
Do you have any suggestions for future articles? You can post on this TI Life issue, or contact me directly at my e-mail address … firstname.lastname@example.org.