The New Sudoku Players' Forum

[gurth]

MY AESTHETICS OF SUDOKU

The concept of the PEARL is my major contribution to this aesthetics. I had often felt dissatisfaction at the way supposedly difficult puzzles could allow one to make an easy start, by immediately placing a single, or even more than one single. This I call an aesthetic dissatisfaction, since I can't really see it as a logical reason to reject any sudoku. Nevertheless, on aesthetic grounds, I reject all supposedly difficult sudokus, that is all sudokus which seem to seek a reputation as difficult, which are not "pearls". I don't mean I reject them entirely : only to a slight degree.

Time to define "pearl".
(1) It must be difficult to solve any cell at the start. Whether or not there is say an easy elimination of candidates by "locked candidates" at the start is quite immaterial, what is material is the total amount of difficulty that must be overcome before the first cell can be placed.

(2) It is absolutely essential to disqualify any sudoku from the title of "pearl" if it contains any redundant clue whatsoever. It this rule were not strictly enforced, then any number of "ARTIFICIAL" pearls could be created, thus driving the market value down, by simply solving ANY difficult sudoku up to the point where maximum difficulty is encountered, and then palming that semi-completed and indigestible mess off as a "sudoku".

(3) While on the subject of redundant clues, let me say at once that I regard any sudoku containing any redundant clue as an aesthetic monstrosity, and even a logical and cultural monstrosity. Not to mention a psychological crime : an insult to the intelligence of the solver.

This practice is sometimes condoned in the interests of preserving "symmetry". I concede that symmetry can have great theoretical interest, which may even legitimately supersede aesthetic importance (aesthetics is not everything!) But I deny any such symmetrical puzzle the title of "pearl" if there are any redundant clues. At most, it can be regarded as a "flawed" pearl.

Symmetry itself has no aesthetic value. The Mona Lisa is not symmetrical, nor is any other work of art worthy of the name. Any form of perfection is fatal aesthetically. The Japanese use the term "zbumi" for the necessary imperfection that any work of art must contain, and which must be deliberately inserted by the artist if it does not already exist.

[Smythe Dakota]

.... I regard any sudoku containing any redundant clue as an aesthetic monstrosity ....

Some solvers use uniqueness arguments to solve puzzles, e.g. they figure out that a 3 can't go in r2c5 because then (somehow) there would be a dual solution involving 4's and 8's in r2c7, r2c9, r6c7, r6c9. Such uniqueness logic relies on the puzzle composer's assurance that the solution is unique.

If, similarly, you were attempting to solve a puzzle which the composer assured you was a "pearl" (no redundant clues), would you use non-redundancy to help you solve it? For example, if you could somehow establish that a 4 in r7c6 would make the 8 given in r3c5 redundant, would you then remove 4 as a candidate in r7c6?

As you can see, the pearl quality of a puzzle could actually make it easier to solve, because it enables "pearl" logic, which is normally not safe to assume. Thus the pearl, an apparently perfect work of art, could contain the seeds of its own destruction. Maybe that's why artists insist that a perfect work of art must include an imperfection.

Ah, the Tao of Sudoku ....

Bill Smythe

[gurth]

Dakota Bill, (if I may call you that):

That's a very interesting idea that had never occurred to me. Yes, if non-redundancy was guaranteed, you could use it! Why not? That is if you could think of a way how.

Wouldn't that be nice, to allow a whole new breed of techniques to spring into being! I propose that composers, from now on, try to maintain non-redundancy in their puzzles, and guarantee it. Down with spurious symmetry, as opposed to mathematically significant symmetry. For the desire to maintain symmetry is the main reason for the existence of all these bloated, padded sudokus. I call them chocolate-box kitsh!

"Minimal" and "non-redundant" are not very attractive words, but "pearl" is reserved for puzzles difficult to start. They belong to the set of minimal puzzles, but only form a small percentage of that set.

I feel at a loss to suggest an apt term. Can anybody come up with a good idea here? Meanwhile, I suggest we stick with "minimal".

Unfortunately I published my pearl G910 simultaneously with this posting, assuming everybody was going to read both, but of course that didn't happen, so many people who read my G910 posting only got entirely the wrong idea of what I mean by pearl, as I didn't define it there also.

They also thought that a pearl had to be at least harder than the Simple Sodoku's set of techniques, but that is not so. Small pearls can also exist and give satisfaction, as long as the difficulty of the first cell is greater than what I call elementary technique, which is subsets and locked candidates.

But some also thought that you could take a puzzle starting with say a single, and then add that single in as a clue to create a pearl! That is precisely what I call an ARTIFICIAL pearl or a FLAWED pearl. Of little if any value. Non-redundancy is essential to protect the "market" value, which depends on scarcity.

[Smythe Dakota]

.... For example, if you could somehow establish that a 4 in r7c6 would make the 8 given in r3c5 redundant, would you then remove 4 as a candidate in r7c6? ....

.... That's a very interesting idea that had never occurred to me. .... Wouldn't that be nice, to allow a whole new breed of techniques to spring into being! ....

This could get tricky. In the above example, what if (for example) the removal of the 4 as a candidate in r7c6 now makes (somehow) the 4 given in r4c8 redundant? Then you would have to remove the given 4 to maintain minimality. But what if such removal destroys (somehow) the original argument, so that it is no longer possible to conclude that 4 is not a candidate in r7c6?

We may have stumbled onto (or be able to construct) a situation where the existence of a given makes the puzzle non-minimal, while its absence results in a dual solution.

Is this the Russell Paradox of minimal Sudoku?

Bill Smythe