Here are a few of the distinct situations she encounters:
(A) She narrows down the possibilities of three cells as shown:
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+------------+------------+------------+
| 12 12 13 | . . . | . . . |
She deduces:
1) r1c1=1 => r1c3=3
2) r1c1=2 => r1c2=1 => r1c3=3
3) Therefore, r1c3=3
[Note: When she writes "A => B", the "=>" means "implies that". She could have just as easily written: "If A, then B." Same thing.]
(B) She narrows down the possibilities of four cells as shown:
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+------------+------------+------------+
| 12 23 13 | 14 . . | . . . |
She deduces:
1) r1c1=1 => r1c4=4
2) r1c1=2 => r1c2=3 => r1c3=1 => r1c4=4
3) Therefore, r1c4=4
(C) She narrows down the possibilities of 7 cells as shown:
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+------------+------------+------------+
| 12 23 34 | 45 56 16 | 17 . . |
She deduces:
1) r1c1=1 => r1c7=7
2) r1c1=2 => r1c2=3 = > ... => r1c6=1 => r1c7=7
3) Therefore, r1c7=7
[Note: if any one of the first 6 cells were removed, she could make no deduction about r1c7.]
(D) She narrows down the possibilities of 4 cells as shown:
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+------------+------------+------------+
| 12 . . | 13 . . | . . . |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
+------------+------------+------------+
| 23 . . | 13 . . | . . . |
She deduces:
1) r1c1=1 => r1c4=3 => r4c4=1
2) r1c1=2 => r4c1=3 => r4c4=1
3) Therefore, r4c4=1
(E) She narrows down the possibilities of 4 cells as shown:
- Code: Select all
+------------+------------+------------+
| 12 . . | 13 . . | . . . |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
+------------+------------+------------+
| 23 . . | 34 . . | . . . |
She deduces:
1) r1c1=1 => r1c4=3 => r4c4=4
2) r1c1=2 => r4c1=3 => r4c4=4
3) Therefore, r4c4=4 [corrected -- thanks Doyle]
[Note: This case is distinct from (D) in than no two cells have identical possibilities. It is however essentially the same situation as (B) except that the cells with possibilities (23) and (13) are not in the same group and do not directly connect to each other.]
(F) She narrows down the possibilities of 8 cells as shown:
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+------------+------------+------------+
| 12 . 23 | 34 . . | . . . |
| . . . | 45 . . | . . . |
| 14 . . | . . . | . . . |
+------------+------------+------------+
| 24 25 . | 56 . . | . . . |
She deduces:
1) r1c1=1 => r3c1=4 => r4c1=2 => r4c2=5 => r4c4=6
2) r1c1=2 => r1c3=3 => r1c4=4 => r2c4=5 => r4c4=6
3) Therefore, r4c4=6
(G) She narrows down the possibilities of 8 cells as shown:
- Code: Select all
+------------+------------+------------+
| 12 . . | 34 . . | 23 . . |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
+------------+------------+------------+
| 24 . . | . . . | . . . |
| . . . | . . . | . . . |
| . . 25 | 56 . . | . . . |
+------------+------------+------------+
| . . . | . . . | . . . |
| . . . | 45 . . | . . . |
| 14 . . | . . . | . . . |
+------------+------------+------------+
She deduces:
1) r1c1=1 => r9c1=4 => r4c1=2 => r6c3=5 => r6c4=6
2) r1c1=2 => r1c7=3 => r1c4=4 => r8c4=5 => r6c4=6
3) Therefore, r6c4=6
===========================================================
Note: The logical connections between the cells are *identical* in (F) and (G). In each case, no two cells have the same two possibilities. If she refers to cells by their remaining possibilities, she can write the following for
BOTH examples:
1) (12)=1 => (14v=4 => (24)=2 => (25)=5 => (56)=6
2) (12)=2 => (23)=3 => (34)=4 => (45)=5 => (56)=6
3) Therefore, (56)=6
===========================================================
(H) She eliminates the possibility of the digit '1' from all but the 4 cells a, b, c and d in rows 1 and 4.
There is one other cell that might contain '1' in columns 2 and 5:
- Code: Select all
+------------+------------+------------+
| . a . | . b . | . . . | <-- only two spots for '1' in this row
| . . . | . . . | . . . |
| . . . | . . . | . . . |
+------------+------------+------------+
| . c . | . d . | . . . | <-- only two spots for '1' in this row
| . . . | . . . | . . . |
| . . . | . 13 . | . . . |
+------------+------------+------------+
| . . . | . . . | . . . |
| . . . | . . . | . . . |
| . 12 . | . . . | . . . |
+------------+------------+------------+
She deduces:
1) a=1 => r9c2=2
2) a/1 => b=1 => d/=1 => c=1 => r9c2=2
3) Therefore, r9c2=2
4) a=1 => c/=1 => d=1 => r6c5=3
5) a/=1 => b=1 => r6c5=3
6) Therefore, r6c5=3
[Here she is using "/=" to mean "is not equal to". Though she's never heard of an 'x-wing', she does notice that this formation is similar to (A).]
(I) She eliminates the possibility of the digit '1' from all but the 6 cells a through f in rows 1, 4 and 7.
There is one other cell that might contain '1' in column 2.
- Code: Select all
+------------+------------+------------+
| . a . | . b . | . . . | <-- only two spots for '1' in this row
| . . . | . . . | . . . |
| . . . | . . . | . . . |
+------------+------------+------------+
| . c . | . . . | . d . | <-- only two spots for '1' in this row
| . . . | . . . | . . . |
| . . . | . . . | . . . |
+------------+------------+------------+
| . . . | . e . | . f . | <-- only two spots for '1' in this row
| . . . | . . . | . . . |
| . 12 . | . . . | . . . |
+------------+------------+------------+
She deduces:
1) a=1 => r9c2=2
2) a/=1 => b=1 => e/1 => f=1 => d/=1 => c=1 => r9c2=2
3) Therefore, r9c2=2.
[She's never heard of a 'swordfish', she does notice how this is similar to (B).]
(J) She eliminates the possibility of the digit '1' from all but the 8 lettered cells in rows the noted
rows.
There is one other cell that might contain '1' in column 2.
- Code: Select all
+------------+------------+------------+
| . a . | . b . | . . . | <-- only two spots for '1' in this row
| . . . | . . . | . . . |
| . . . | . . . | . . . |
+------------+------------+------------+
| . c . | . . . | . d . | <-- only two spots for '1' in this row
| . . . | . . . | . . . |
| . . . | . e f | . . . | <-- only two spots for '1' in this row
+------------+------------+------------+
| . . . | . . g | . h . | <-- only two spots for '1' in this row
| . . . | . . . | . . . |
| . 12 . | . . . | . . . |
+------------+------------+------------+
She deduces:
1) a=1 => r9c2=2
2) a/=1 => b=1 => e/1 => f=1 => g/=1 => h=1=> d/=1 => c=1 => r9c2=2
3) Therefore, r9c2=2.
(K) In this case, all the lettered cells must be either 1 or 2, while r9c8 can be 1, 2 or 3:
- Code: Select all
+------------+------------+------------+
| . . . | . . . | . . . |
| . . . | . g . | h . . |
| . . e | f . . | . . . |
+------------+------------+------------+
| . . d | . . . | . . . |
| . . . | . . j | i . . |
| . c . | . . . | . . . |
+------------+------------+------------+
| . . . | . . k | . l . |
| . b . | . . . | . . . |
| a . . | . . . | . 123 . |
+------------+------------+------------+
She deduces:
1) a=1 => b=2 => ... k=1 => l=2
2) a=2 => b=1 => ... k=2 => l=1
3) Therefore, r9c8=3
(L) In this case, the same cells as in the above puzzle each of two possibilities:
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+------------+------------+------------+
| . . . | . . . | . . . |
| . . . | . 78 . | 89 . . |
| . . 56 | 67 . . | . . . |
+------------+------------+------------+
| . . 45 | . . . | . . . |
| . . . | . . 12 | 91 . . |
| . 34 . | . . . | . . . |
+------------+------------+------------+
| . . . | . . 23 | . 13 . |
| . 23 . | . . . | . . . |
| 12 . . | . . . | . 19 . |
+------------+------------+------------+
She deduces:
1) r9c1=1 => r9c8=9
2) r9c1=2 => r8c2=3 ... [follow same logical path as in (K)] ... => r7c8=1 => r9c8=9
Each puzzle seemed brought new situations. She was able to build on what she learned solving the easier ones to use more complex logic to solve the harder ones. Though she often went back and solved them again, she NEVER found that a newer, more complex method replaced one that she had earlier found, even in a specific instance. She would use the methods she found easier until she could go no further, try something harder to push forward, then go back to the easier ones again.
She eventually worked on puzzle in which she was unable to reduce the possibilities for *any* of the cells to less than three -- but she was still able to crack them using more ingenious logical methods that I would like to explain here but there is no room in the margin.
Still, in the end, there was one puzzle that was so hard that she could make very little headway. Though she suspected it was very likely that there were methods yet undiscovered that would it to be simplified, and with experience, study and practice she would uncover them, for now, it was beyond her. She instead choose to solve this last one by what most people refer to as "trial and error" or "bifurcation" but is actually "proof by contradiction" or "Reducto ad absurdum".
She made an assumption and followed it until she found a contradiction, thereby enabling her to eliminate her assumption from the possibilities. While solving this one, she was forced to use *all* the skills she had learned early - in *addition* to making these assumptions. She enjoyed solving this one quite a bit. It wouldn't occur to her to use this tactic unless and until she could find no other way -- until she had *proved* to herself there was no other way.
When she had solved the last of the puzzles -- which was an enjoyable task for her -- she showed her solutions along with her methods for solving to several people far more familiar with Sudoku, many of whom are far more educated than she is and obviously very intelligent. She is told many strange things, conflicting things, including, but not limited to:
-- She "guessed" the answer in some of the puzzles.
-- She used "trail and error" in some of the puzzles.
-- She used methods that were "not elegant".
-- No one enjoys solving puzzles using some of the methods she used.
-- Though she solved them, some of the puzzles are "invalid" -- as no human could possibly solve them.
-- Many of the puzzles she solved are "not solvable by logic".
-- She "cheated herself" of the joy of solving some of the puzzle by "real logic".
-- All types of logic puzzles -- if correctly formed -- can always be solved by "just logic" and never
require methods analogous to some that she has used.
-- She used "inductive logic". (A claim she knows simultaneously false and pointless.)
Which puzzles were "invalid" and/or "not solvable by logic" varied from person to person. Which methods were "guessing", "not elegant" or "fun" varied from person to person. Which puzzles were "beyond
human ability" unless you drew lots of lines and charts, which is there for "not fun" varied from person to person.
Oddly, how subjectively difficult or how long it took to solve any individual puzzle seemed only loosely correlated to whether or not the puzzle was labeled "invalid" or not -- some of the "invalid" puzzles were actually easy.
Though this woman is imaginary, nonetheless, she came to me and complained about these ironically illogical statements about logic. She wondered what reasoning was behind trying separate into categories things that were so obviously one and the same, different only in degree, a continum of logic from simpler to more comlex. Would one differentiate 2*2=4 from 123*456=56088 simply because we *know* the first but must *calculate* the second? Does it change if we memorize that second product? Or if we forget the first for that matter? Is it more fun or elegant to fill in 3x2=6 than to perform multi-digit multiplication? Am I the only person that enjoys trying to muliply two 5 digit numbers in my head while stuck in traffic?
I asked "So what? You can solve as you like. What does it matter if so many others have it wrong?"
She agreed except for one thing. Some of the people who create these puzzles, who have decided arbitrarily what is and isn't a "valid" Sudoku, are treated as "authorities" -- by the press, by solvers, by other puzzle creators, many or most of which seem to be following suit, declaring certain constructions as "non-puzzles", "not solvable by logic", etc., though again, where each of them will draw the line is arbitrary.
