Sudoku Hole

Liang's Sudoku and odd hole theorem  Definition 1: sudoku nine nine palace any a line of three lines of three two digits x1 and d1, exist in the same row of two other nine palace a nine palace of three lines, or any nine palace in the same line of two Numbers x1, d1, can be in the same row of two other nine palace in three lines of three digits find two, we say the same line of three x1, d2 two numbers is twin T brother, the rest of a number g1 called a soliton. Nature 1: three x1, d1 in the nine palaces, must be occupied by the three palaces, there is no two palaces, no palace, the same row of no x1, d1 in the twin T.  Property 2: the same row of three nine houses not only meet property 1, x1, d1, there must be x2, d2 of twin T, and x3, d3 of twin T, and the corresponding g2, g3. x1d1，g1 ； x2d2，g2； x3d3，g3。 We can also define the second row of the corresponding twin T brothers, as well as the lone son.  x‘1d’1，g‘1 ；  x’2d‘2，g'2； x’3d‘3，g’3。  Similarly, there can be a third row of nine palaces corresponding to:  x”1d”1，g”1 ；  x”2d”2，g”2；  x”3d”3，g”3。  Therefore, there are only three solons in the same nine palace. Such as g1, g2, and g3. Similarly, any one of the three columns of three two digits z1 and m1, exist in the same column two other nine palace a nine palace of three columns, or any nine palace in the same column of two digits z1, m1, can be in the same two nine palace a nine palace three columns find two three digits, we say the same column three digits z1, m2 two numbers are twin T sister, the rest of a number G1 is called an orphan.  z1m1，z2m2，z3m3，  G1 ， G2， G3；  The second column nine palace  z‘1m’1，z‘2m’2，z‘3m’3， G‘1 ， G’2， G‘3；  The third column nine palace  z”1m”2，z”2m”2，z”3m”3，  G”1， G”2， G”3；  Therefore, there are only three orphan women in the same nine palaces. For example, G1, G2, and G3.  Definition 2: in the same nine palace, if G and G are the same position, that is, the same number, it is called a single point, or cave D. The distribution of the nine houses is as follows:  D11，D12，D13；  D21，D22，D23；  D31，D32，D33；  Definition 3: If all nine or nine palaces have nine individual dots, we call them nine laps, or nine holes. According to definition 2, and definition 3, we have Liang's number theorem 1: any Suddoku in a nine palace will not be more than two. to be verified. Liang's number odd hole theorem 2: the number of all single points in any sudoku does not exceed its maximum value of 12. to be verified.  Liang's number odd hole theorem 3: the number of all single points in any sudoku is not less than its minimum value of 6. to be verified. 

The text you provided describes a theory about Sudoku puzzles called "Liang's Sudoku and Odd Hole Theorem." Here's a breakdown of the key points:
Terminology:

• Twin T Brothers/Sisters: Two digits appearing together in the same row or column of three different 3x3 boxes (palaces).

• Soliton/Orphan: The single remaining digit in a row or column with Twin T brothers/sisters.

• Single Point/Cave D: When a Soliton/Orphan occupies the same position (row and column) in different palaces.

• Nine Laps/Nine Holes: A Sudoku puzzle where all nine palaces contain nine unique Single Points/Cave Ds.

Theorems (Unverified):

• Liang's Number Theorem 1: Any Sudoku puzzle will not have more than two Single Points/Cave Ds in a single palace. (Needs verification)

• Liang's Number Odd Hole Theorem 2: The total number of Single Points/Cave Ds in a Sudoku puzzle will not exceed 12. (Needs verification)

• Liang's Number Odd Hole Theorem 3: The total number of Single Points/Cave Ds in a Sudoku puzzle will not be less than 6. (Needs verification)

Overall, the text proposes a theory about the distribution of Single Points/Cave Ds in Sudoku puzzles. The theory suggests limitations on the number of Single Points/Cave Ds within a single palace and the entire puzzle.
Points to Consider:

• The text focuses on specific terminology and definitions, so some context might be missing.

• The provided theorems haven't been verified and might require further investigation or mathematical proof.

• It's unclear how this theory relates to solving Sudoku puzzles or its overall impact on the game.

Further Exploration:

• If you're interested in the theory, you could try searching for "Liang's Sudoku and Odd Hole Theorem" online to see if there are any existing studies or discussions about it.

• You could also try applying the theory to different Sudoku puzzles and see if it holds true in all cases.

• Researching existing Sudoku solving techniques might be more helpful for actually solving Sudoku puzzles.

I hope this explanation clarifies the concepts presented in the text. Feel free to ask if you have any further questions or need clarification on specific points.