Liang Cube Sudoku:
Centurial Puzzle

Haisheng Liang

Tagle of Contents

Preface 3

Chapter 1 Finding the Diagonal of Sudoku 14

Chapter 2 Sudoku with Three Field Segmentation Rows or Columns 16

Chapter 3 Six Colors Like Rubik's Cube 21

Chapter 4: Non diagonal "Clockwise Position" 25

Appendix 31

Preface

On April Fool's Day 2024, a big international joke can finally be played. The two most famous, popular, and interesting intellectual toys of the 20th century, Rubik's Cube and Sudoku, were organically integrated by Liangkuai in the 21st century, becoming the most fun intellectual toys of the 21st century. A groundbreaking and disruptive reconstruction!
Actually, you know my work, but you don't know me. Who am I? I am the inventor of the Love Gesture. Thirty years ago, in 1993, at Sony Music's art exhibition, the Love Gesture photos I created were first featured in the art exhibition's portfolio. After the evolution of different versions of Love Gesture, they have become popular worldwide.
My great discoveries on April Fool's Day in the the Year of the Loong are as follows.
Problem requirement: Rubik's Cube 3x3x3 has 27 small particles, divided into three parts: A, B, and C. Each part has nine small particles, which can be connected together and cannot fly away. All three blocks cannot be in the form of 3x3 nine palace pieces. Each grain is marked with the same nine digits of 1-9 as Sudoku.
Moreover, this "Rubik's Cube" can be divided into three layers along the X-axis direction, with nine small particles in each layer, all of which are the small Nine Palaces of Sudoku. Along the Y direction, this Rubik's Cube can also be divided into three layers, each with nine grains, all of which are the small Nine Palaces of Sudoku.
Obviously, this allows for the coloring of a Rubik's Cube, which only lacks four faces on the opposite side. Each face can be painted with the same color for nine grains, and the nine grains on one face all have different numbers of 1-9.
In addition, the remaining fifth face can have at most one small grain number repeated and can be marked with the color on the opposite side.
On the last sixth side, there can be up to two small particles repeating two pairs of numbers, and the two small particles can be marked with colors on the opposite side.
The "Rubik's Cube" that meets the above requirements is abbreviated as Liang Cube, and its full name is Liang Cube Sudoku.
The birth of this Liang Cube Sudoku happened 42 years ago, when college student Liang Haisheng first came into contact with the Rubik's Cube in 1982.
In 1986, the Japanese puzzle journal Nikoli began publishing Sudoku games.
In the same year, due to its resemblance to Rubik's Cube, Liang Haisheng began to think about a square movement and space filling cube game called "Magic Stinger" by the Fun Mathematics column of the Scientific American Journal. In 1987, he invented a mortise and tenon block with six faces and three axes that are coaxial and parallel to each other, and opposite axes that are perpendicular to each other. He won an award and applied for it in Shanghai, and was granted dual patents for utility model and appearance design by the Chinese Patent Office in 1988.
In 1987, the Shanghai Intelligent Toy Awards also included the "Kirkman Girls Walking Problem" toy, which began researching fun mathematical solutions for the Nine Palace Grid. See.
In 1988, Liang Haisheng came to Japan to study and saw Nikoli's Sudoku for the first time.
Six years have passed since college student Liang Haisheng first encountered Rubik's Cube in 1982.
In 1993, the contour processing 8-way chain code technology (3-row format) based on image processing technology was subversively applied by Liang Haisheng to represent the broken lines of digital columns. Contemporary art works were presented at the second AAA Contemporary Art Exhibition held by Sony Music Entertainment Co., Ltd.
In 1994, Liang Haisheng invented a 1.5-dimensional barcode and applied for a Japanese patent. Four years later, in 1998, the American Auto I. D. Introduction to the front page headlines of News. The global circulation is 100000 copies. Eight years later, in 2002, he won the Japan Open Business Plan Contest award from Kenichi Ohmae.
In 1995, the 26 directional representation concept of the 27 block structure of a three line computer keyboard was presented at the annual conference of the Japan Information Processing Society's Image and CAD. Thirteen years have passed.
In 1997, the magic square of the 4x4 determinant was published online.
In 2005, the second year after Liang Haisheng returned to China from Japan, Sudoku became popular worldwide due to the reprint in The Times.
In 2011, Liang Haisheng's groundbreaking invention, the rectangular three-dimensional origami, ceased to consider new inventions and creations. Instead, as an artist, he focused on creating related art exhibitions and art education, as well as offering a series of continuous three-dimensional origami courses in primary schools.
Since then, I have been working as an artist in art and early childhood education. Organize domestic and international art exhibitions, origami exchanges, and teaching activities at the Italian Paper Art Biennale, the French Origami Association, and the Japan Origami Education Mathematics Research Association.
This is a big discovery article that needs improvement. After five years of the pandemic, I was happy and a little excited to see Japanese cherry blossoms again. On April Fool's Day, Amazon editors supported several books with numerical columns and works in various languages such as the trisection cube. During the KDP inspection, I was very excited as an artist when I first saw ten different colors of Amazon stripes.
Thank you very much to Amazon Japan's publishing department for editing. With this, the deep desire to make a breakthrough in the field of Sudoku (disruptive and epoch-making) was inspired by the fact that in 1995, the digital line represented the direction of 27 blocks and 26 directions in space. I believe it will bring infinite surprises and fun to millions of Rubik's Cube puzzles and Sudoku fans around the world in a "screaming mode".
At a critical moment when AI made significant progress, Liang Haisheng proposed the Liang's AI law that "AI cannot propose new rules for quantitative logic.".

ASIN  :  B0CXJWQX6Z  Mortise and Tenon Cube Building Blocks；
ASIN  :  B0CNPXMR72 The Number String Graphics Design；
ASIN  :  B0CYCRLXJK  Auspicious Magik Square；
ASIN  :  B0CYQ82BFS   Number String Fantasia；
ASIN  :  B0CZDTPPP7  How to Trisect Cube with Polyhedron；
ASIN  :  B0CTHSVHWQ  Romance of the Three Kingdoms: Liang Cube；

Liang Haisheng

Freedom Hill, Tokyo, Japan
April 1, 2024
Chapter 1 Finding the Diagonal of Sudoku

The definition of Xiaojiugong is in line with the definition of Sudoku. The minimum 3x3 grid in the nine rows and nine columns of Sudoku is arranged with nine numbers ranging from 1 to 9.

The definition of the Big Nine Palaces is that a Sudoku 9x9 grid is divided into 3x3 "Little Nine Palaces". Each "Little Nine Palaces" includes

The nine elements of the corresponding 3x3 matrix are all pairwise different.

Liang Haisheng's Sudoku Theorem 1: Each Sudoku has at least one diagonal in the row or column of the Big Nine Palace.

Liang Haisheng's Sudoku Theorem 10: Each Sudoku has a maximum of three diagonal lines in a row or column of the Big Nine Palace.

Sudoku Theorem 11: The basic nine houses where the midpoint of a diagonal is located must be outside the center of the three rows of Sudoku nine houses it is in.

Chapter 2 Sudoku with Three Field Segmentation Rows or Columns

Sudoku Theorem 2: A 3x3 cube S is located in the first quadrant of the XYZ axis and has an angle at the origin. Obviously, a unit long cube has 3x3x3=27 small pieces.
Assuming a 3x3 cube S is divided into three blocks A, B, and C, each with nine small particles that can be connected together but cannot fly away. None of the three blocks can be in the shape of the largest 3x3 nine palace piece. Each grain is marked with the same nine digits of 1-9 as Sudoku. So define S as Liang Cube.
So, for the following conditions, the Liang Cube can all meet:
1. This "Liang Cube" can be divided into three layers along the X-axis direction, with nine small particles in each layer, all of which are the small Nine Palaces of Sudoku.
2. Along the Y direction, this "Liang Cube" can also be divided into three layers, each with nine small particles, all of which are the small Nine Palaces of Sudoku.
Obviously, this "Liang Cube" can be used to paint a Rubik's Cube, with only two parallel and opposite faces, four colors and four faces. Each face can be painted with the same color for the nine small Nine Palace pieces, each with different numbers ranging from 1 to 9.

Sudoku Theorem 20: The S of the Liang Cube can be divided into three blocks, A, B, and C, which comply with the definition in Sudoku Theorem 2 above. These three blocks of A, B, and C also have at least one method in space that corresponds to each block with nine small particles and nine different numbers, with a maximum of N methods.

Chapter 3 Six Colors Like Rubik's Cube

Sudoku Theorem 3: The Liang Cube only satisfies the XYZ triad, where there are one axis, three types (three layers), and two axis, six types (six layers) of the Sudoku Little Nine Palace. There is no "perfect Liang Cube" that satisfies all three axis, nine types (nine layers).

Chapter 4: Non diagonal "Clockwise Position"

If we understand the diagonal of Chapter 2 as the hour and minute hands of a clock, Chapter 3 is like a Rubik's cube with six colors and six dots, then what would the three positions of the same number on the non diagonal look like

Sudoku Theorem 12: Definition: If one of the elements with the same number in a row of three Sudoku Nine Palaces is located in the center, it is called a "clockwise" C.
Conclusion: The case R where the other two identical numbers of the three identical numbers form 90 degrees with the same number in the center position is limited to the position where both numbers are in the "little nine houses" of Sudoku. That is to say, there is no position where both numbers are located on either side of the four sides of the Little Nine Palace.

Several types of three digit positions in Sudoku.

1 Diagonal/Shaped