The New Sudoku Players' Forum

[gurth]

PEARLS, RUBIES, GARNETS, EMERALDS, ETC.

For quick reference, definitions of above terms :

PEARL :

A Pearl is a minimal sudoku where the first solvable cell is difficult to solve. The more difficult, the greater the pearl and its value.

A small pearl would require more than elementary technique, i.e. more than singles, pairs, trips, quads and locked candidates.

A large pearl would require more than SSTS, ie more than elementary technique plus Swordfish, X-Wing, XY-Wing, 2-colour and 4-colour chains.

If the puzzle is not minimal, i.e. contains any redundant clue/s, then the pearl is a fake, by definition. This definition is necessary to protect the scarcity value of pearls, otherwise any non-pearl could be made into a pearl by inserting additional clues until a point of difficulty was reached.

Two great examples of pearls : Ocean's Pearl Brooch and Claudia's Star of Arabia.


RUBY :

A Ruby is, strictly speaking, a non-conjugate candidate x in a sudoku, whose removal from a cell at the very start of the solution leads to a contradiction and whose placement leads to solution, both processes requiring no more techniques than SSTS, and preferably even a simpler set of techniques.

The simpler the techniques required, the greater the Ruby.

Non-conjugacy requires that there be more than 2 candidates in the Ruby Cell, and that the Ruby not be a member of a conjugate pair of x in row, column or box.

The greater the non-conjugacy, i.e. the more candidates in the Ruby cell, and the more other x-candidates in the row, column and box, the greater the Ruby.

The non-conjugacy rule increases the scarcity value of rubies, and so makes them deserving jewels.

Finally, the greater the overall difficulty of the puzzle, the greater the Ruby. A Ruby requires a puzzle that requires more than elementary techniques to solve. Note that the Ruby CONTAINS only simple techniques, but is not ITSELF a simple technique but the start of a Forcing Net or other more complex technique.


GARNET :

A Garnet is the same as a Ruby, except that it is not present at the very start of the solution, but arises later. Garnets are of less value than rubies, because they occur far more frequently.


EMERALD :

An Emerald is defined as a sudoku with clues which are symmetrical in their natures and mutual relationships as well as in their positions.

A prime example of an Emerald is Tso's Puzzle, made famous by Carcul in his controversial "Riddle of Sho".


TURQUOISE:

A Turquoise is defined as a minimal sudoku where it is difficult to eliminate the first candidate. It can be measured as the SE rating of the first step, required to eliminate the first candidate.


JADE:

Somewhat loosely defined as a sudoku which starts off easy and becomes hard later. The ultimate Jade would be the puzzle starting with the most singles, followed by an extreme jump in SE rating for the next step.


re: Precious Stones Guidebook

Tarek,
I have opened THIS topic "PEARLS, RUBIES, GARNETS, EMERALDS, ETC." giving definitions of these terms, for which I am responsible.

DIAMONDS are not my province, but rather udosuk's I think. Refer to him regarding diamonds, please.

udosuk,
and anyone else, please feel free to use the new topic to add your own jewelry definitions. Then everything will stay together for ready reference.

For that matter, why confine yourself to gemstones? Anything goes! Maybe flowers... Or else proper names such as "Mona Lisa" ?

[udosuk]

Along gurth's line of descriptions...

DIAMOND:

A Diamond is a valid sudoku puzzle with no backdoor cells (under a certain technique set). A backdoor cell is a cell which if assumed the correct value (one that matches the unique solution) will allow the puzzle to be solved logically under the technique set in scope. (This is half of the requirement of a Ruby cell.)

The smallest/most common Diamonds are those with singles (hidden/naked) as the technique set. An example is this one by Ocean:

Code: Select all

1.......2
.3..4..5.
...6.7...
..4...6..
.7..8..4.
..9...8..
...9.8...
.2..7..1.
5.......3

Which is btw a Pearl too (and a tough one, coined by gurth as "OCEAN'S PEARL BROOCH")...

A large Diamond is one which has no backdoor cells under SSTS, and the only known one so far is (by Ocean too):

Code: Select all

.....1..2
.3..4..5.
6..2.....
..5.....3
.7..8..4.
2.....9..
9....4...
....5..7.
..41..6..

SAPPHIRE:

A Sapphire is a valid sudoku puzzle that is not solvable under propositions (1-level T&E) using a certain technique set.

"Proposition (1-level T&E)" means we could allow assuming a certain candidate in each cell, and if that assumption leads to contradiction, we eliminate that candidate from that cell and resume the solving of the puzzle. If it turns out to be the correct value that leads to the final solution, we have the puzzle solved (making that cell a diamond cell). If the result is neither of them, we leave the candidate list of that cell unchanged (with the tested candidate staying there) and resume the solving of the puzzle.

If we cannot progress to the solution using the aforementioned procedure, i.e. get stuck, then the puzzle is a Sapphire. A Sapphire must be a Diamond under the same technique set (because it must not have any backdoor cells). It's highly likely that no Sapphires exist under SSTS, and so far there is only one Sapphire found under the technique set "Singles+Locked Candidates+Naked Pairs":

Code: Select all

1.......2
.3..4..5.
..6...7..
...1.3...
.8..7..4.
...4.6...
..2...6..
.5..3..8.
9.......1

Which is found by gsf by testing a bunch of Ocean's puzzles... According to him there are 4 Sapphires under "Singles+Locked Candidates" (including the one above), and 27 Sapphires under "Singles" only, sampling on 55 of Ocean's hardest puzzles...

[ravel]

A large Diamond is one which has no backdoor cells under SSTS, and the only known one so far is (by Ocean too) ...

Are there SSTS backdoor cells in Tarek's 20-step puzzle (with the basics i have implemented i did not find one) ?

[udosuk]

Are there SSTS backdoor cells in Tarek's 20-step puzzle (with the basics i have implemented i did not find one) ?

Under SSTS, r1c3 & r5c7 are the only backdoor cells (both require multiple colors), so it's not a "large Diamond"... But a great "medium" one...

Being a Diamond/Sapphire doesn't seem to correlate too much on the difficulty rating you or SE adopt... gsf's rating is more related I suppose...

[RW]

Here's probably the softest version of a diamond, not very common. It's not solvable by singles, locked candidates and subsets, neither has it got a backdoor under these techniques. However, it is solvable all the way under SSTS.

Code: Select all

 *-----------*
 |9..|.1.|2..|
 |...|.83|...|
 |1..|6..|.8.|
 |---+---+---|
 |5..|...|6..|
 |69.|...|.18|
 |..8|...|.2.|
 |---+---+---|
 |.5.|..6|..3|
 |...|7..|...|
 |..2|5..|.71|
 *-----------*

RW

[tarek]

Nice Gurth.....

I suggest changing the thread's name to The precious stone guide book...........

Pearls are very interesting.........I remeber Vidarino posting one of his monsters (which was in the diagonal pattern) that was a pearl....

The pearl describes a difficult 1st hurdle........I think another gemstone should be reserved for overall difficult puzzles (I don't know TOPAZ or TIGER EYE), then you could combine them. A PEARLY TIGER EYE should be the ultimate.......

tarek

[gurth]

udosuk, I like your Diamond and Sapphire definitions. What do you suggest for overall difficult puzzles? Tarek suggests Tiger Eye or Topaz.
I don't seem to have any ideas of my own here.

tarek, Thanks for your suggestions. One reason why I hesitated to call this topic "Precious Stones Guidebook" is that I didn't want to exclude other precious items, such as a Gold Crown, for example.

Is a pearl really a stone ? I'm not too sure.

[Karlson]

Pearls are formed inside the shell of certain bivalve mollusks. As a response to an irritating parasite inside its shell, the mollusk will deposit layers of calcium carbonate (CaCO3) in the form of the minerals aragonite or calcite (both crystalline forms of calcium carbonate) held together by an organic horn-like compound called conchiolin. This combination of calcium carbonate and conchiolin is called nacre, or as most know it, mother-of-pearl. The idea that a grain of sand acts as a "seed" for the pearl is a myth.

Source: wikipedia.org

[udosuk]

udosuk, I like your Diamond and Sapphire definitions. What do you suggest for overall difficult puzzles? Tarek suggests Tiger Eye or Topaz.
I don't seem to have any ideas of my own here.

"Overall difficult puzzles" is so hard to qualify... Where do we draw the line there? All these definitions here have well-defined properties... If we thought we have the "most difficult sudoku puzzle" today and called it the "Tiger Eye" or so then what if we find a more difficult one tomorrow?

Perhaps we use different precious metals to tag the puzzles according to their SE-rating... Like Platinum for 9.9+ puzzles, Gold for 9.5+ etc...

This link gives you some more precious stones to think about... Some we haven't used include Agate, Jade, Jasper, Opal (world basketball champion! ), Topaz, Turquoise, and Amethyst (used to be one of the "cardinal gems" alongside Diamond, Ruby, Sapphire, Emerald but lost its place)...

As for "organic gems" like Pearl, other examples are Amber, Coral, Jet, Nacre ("mother of pearl") and the now almost-banned Ivory (with one particularly brutal massacre occured recently in Chad)...

[udosuk]

Regarding Diamonds, another notable (medium) one is Ocean's #1/M21/D21 (SE 10.0, 17 steps from ravel) which has the same shape as the only large diamond:

Code: Select all

.....1..2
.1..2..3.
4..5.....
..4.....6
.7..3..1.
8.....9..
5....8...
....1..7.
..64..5..

Under SSTS there is only ONE backdoor cell, r3c5. You don't need multi-colors or xy-wings though, only simple-colors + x-wings + naked quads are required... Contrary to tarek's #1/1 (SE 9.9, 20 steps from ravel) which have 2 backdoor cells but both needing multi-colors...

[tarek]

would this qualify as a diamond?

Code: Select all

 6 . . | . . 9 | 8 . . 
 . 9 . | . 3 . | . 4 . 
 . . 2 | 1 . . | . . 3 
-------+-------+------
 . . 8 | . . . | . . 9 
 . 4 . | . . . | . 3 . 
 2 . . | . . . | 1 . . 
-------+-------+------
 1 . . | . . 8 | 7 . . 
 . . . | . 4 . | . 5 . 
 . . 5 | 7 . . | . . 2

tarek

[Smythe Dakota]

.... A Pearl is a minimal sudoku where the first solvable cell is difficult to solve. ....

If you are solving a Pearl, which you are assured is indeed a Pearl, would the following sort of logic be acceptable?

"r2c5 cannot be a 7, because if it were, the 3 given in r1c6 would be redundant, thus making the puzzle not minimal after all."

-- assuming, of course, that you could somehow develop such logic.

(This is similar to the uniqueness logic that many solvers embrace.)

So, the knowledge that a puzzle is a Pearl might make it easier to solve, right?

Bill Smythe

[RW]

If you are solving a Pearl, which you are assured is indeed a Pearl, would the following sort of logic be acceptable?

"r2c5 cannot be a 7, because if it were, the 3 given in r1c6 would be redundant, thus making the puzzle not minimal after all."
...
So, the knowledge that a puzzle is a Pearl might make it easier to solve, right?

IMO the logic would be acceptable. However, in order to know that the given 3 in r1c6 is redundant you would have to know that all unavoidable sets including r1c6 are covered by other clues. To find all these unavoidable set you would have to know the solution and analyze it for quite a while with a computer. Here Red Ed shows a single clue that hits 82543 unavoidables not hit by any other clues in the puzzle. The other option would be to remove the clue in r1c6, solve the puzzle and find that it still has an unique solution.

Now, say you have a puzzle where r1c6=3 is given and in the solution r2c5<>7. If you assume r2c5=7 and want to show that if that is the right option, then the given 3 in r1c6 is redundant your two options are:

1. Find a solution where r2c5=7 and find all the unavoidable sets and show that all sets are hit by clues other than r1c6.
2. Remove the clue in r1c6, solve the puzzle and find an unique solution.

The first option is impossible, because if the puzzle has an unique solution, then you cannot find a solution where r2c5=7.

For the second option, it is possible that you could find an unique solution by adding r2c5=7 and removing 3 from r1c6, but in that solution r1c6<>3, if the original puzzle had an unique solution.

Hmm.... but suppose the original puzzle isn't unique... yes, the technique can be used here:

Code: Select all

 *-----------*
 |4..|1..|3..|
 |..7|..6|..*|
 |69.|.4.|.2.|
 |---+---+---|
 |1..|...|5..|
 |...|69.|..8|
 |..6|..5|.4.|
 |---+---+---|
 |.8.|..3|...|
 |..5|...|...|
 |2..|8..|.7.|
 *-----------*

Suppose you spilled coffee on your precious pearl and now you can't see what the original clue in r2c9 was... Don't worry, as you know that it was a pearl, you can still solve the puzzle!

[Edit: Can anybody find the even easier way to solve this unique puzzle that is known to be a pearl? ]

RW

[udosuk]

would this qualify as a diamond?

Code: Select all

 6 . . | . . 9 | 8 . . 
 . 9 . | . 3 . | . 4 . 
 . . 2 | 1 . . | . . 3 
-------+-------+------
 . . 8 | . . . | . . 9 
 . 4 . | . . . | . 3 . 
 2 . . | . . . | 1 . . 
-------+-------+------
 1 . . | . . 8 | 7 . . 
 . . . | . 4 . | . 5 . 
 . . 5 | 7 . . | . . 2

Not a large one, as r3c6 is its only backdoor cell under SSTS (with multi-colors needed, but no xy-wing). It beats the current 2nd & 3rd (listed above) so is our new undisputable 1st runner-up... Could be overtaken by one with a single backdoor cell requiring multi-colors+xy-wing though...

About our current sole large Diamond, I wonder if we add one extra technique (such as UR, BUG or ALS) would any backdoor cell emerge... I'm reluctant to bring on xy-chain or forcing net as they're probably too powerful...

[m_b_metcalf]

PEARLS, RUBIES, GARNETS, EMERALDS, ETC.

For quick reference, definitions of above terms :

PEARL :

A Pearl is a minimal sudoku where the first solvable cell is difficult to solve. The more difficult, the greater the pearl and its value.

Just happened to find this today. Not as pretty as Ocean's or Claudia's, but starts off at SE 6.7, then goes higher.

Code: Select all

  3  .  .  8  .  .  .  .  . 
  .  8  .  .  6  .  .  4  .
  .  .  9  .  .  7  .  .  3
  7  .  .  4  .  .  3  .  .
  .  .  .  .  9  .  .  .  .
  .  .  6  .  .  5  .  .  2
  8  .  .  2  .  .  7  .  .
  .  4  .  .  5  .  .  9  .
  .  .  5  .  .  8  .  .  .


Regards,

Mike Metcalf