This week a beautiful symmetrical Sudoku from San Chuan (三川) family is presented for reader to solve.  The answer of this Sudoku is so special that it looks like a self rotating planet or a spinning top if you view the northwest-southeast diagonal as the spinning axis.   This Sudoku puzzle seems difficult to solve at a glance with only eight known clues, but actually it is extremely easy.  However, because of its beauty and super constrained property so I publish it just for the record.  According to its shape, I named it Spinning Top Sudoku (陀螺數魔, tuo luo).

Spinning Top Sudoku is a derivative from another Sudoku which I named Fu Xi Sudoku (伏羲數魔).  Fu Xi Sudoku is derived from a 9X9 Magic number chart constructed with a method that I learned from my junior high school math teacher Ms. Liu Zhen Zhen (劉真真), who gave us a ten minutes quiz every time after her Algebra class.  I should say it was long time ago one day in 1959, after the Algebra class and exam Ms. Liu gave us a very short talk on magic square construction.  It really stirred up my great interest in numbering magic even after almost half a century later I still remember how to construct it.  I used her 81-digits magic number square construction method and renumbered the digits to only nine and got this wonderful Fu Xi Sudoku.  I think it is appropriate to name it Fu Xi.  King Fu Xi is the first human to invent Bagua (八卦) five thousand years ago in China.  Bagua later became Yi Jing (易經, I Ching, or “Book of the Change”).  I have previously published a few related articles about these subjects in this website.

A detailed examination of these two Sudokus, I found they both fulfill the following rules:

1. Nine 9-digits rows.

2. Nine 9-digits columns.

3. Nine 9-digits Sans (三).

4. Nine 9-digits Chuans (川).

5. Nine 9-digits diagonals or pseudo diagonals (view from the angle of a Dieluohan,疊羅漢) parallel to the spinning axis.

6. Nine single-digit diagonal lines each with 9-members perpendicular to the spinning axis (also view from the angle of a Dieluohan).

The above six rules totaled 54 constraints.  In addition to that, Fu Xi Sudoku has two more rules than the Spinning Top Sudoku.  They are:

7. The perpendicular diagonal to the spinning axis should have a sum of all nine digits equal to the sum of the spinning axis diagonal.  This is one constraint.

8. The spinning diagonal should be arranged in descending or ascending order.  This has nine more constraints, since all eight pseudo diagonals also follow this rule.

We therefore get 64 total 9-digits constraints for Fu Xi, already four more constraints than the Jia Zi Sudoku I published last month.

Another interesting phenomenon of San Chuan, Jia Zi , Spinning Top, and Fu Xi Sudokus are they all contain 三 (San) and 川 (Chuan), which resemble the 乾卦 (Qian Gua), the first Gua in Fu Xi Bagua. 

Fu Xi Sudoku requires no clue to get the puzzle solved.  Once the final diagonal rule is declared, a unique solution should be the only answer for a blank 9X9 matrix.

記得在一九五九年,當我還是初二學生的時候,我的代數啟蒙老師劉真真女士,在下課後講解了幾分鐘的魔數方塊的構造方法.現我模仿她的架構,重新組合,發展出一個新的魔數方塊.這個魔數方塊,我稱它為伏羲數魔 (Fu Xi Sudoku), 以用來紀念伏羲帝五千年前首先發現河圖上的神奇數字的組合,及從中演義出來自今仍然全球風行的八卦易經哲理.

伏羲數魔是屬於三川魔術方塊 (San Chuan Sudoku) 的一種.另外我還做了一道相關 的迷題,叫陀螺數魔(Spinning Top Sudoku) 做為本周的迷面.伏羲數魔與陀螺數魔源於一脈,都出於三川魔術.雖然這道迷題只給了八個已知數,看起來好像是很難,其實卻非常容易解的.

伏羲數魔和陀螺數魔主要有兩個限制,既:將由西北到東南方向的大對角線視為轉軸,每一條向它垂直的線上(總共有十五條垂直線)所有的數字都要相同.但是由疊羅漢的角度去看就一共只有九條連續的或是不連續的垂直而同時還包含九個數的直線,這個奇特的對稱性使整個方陣很像一個自轉的陀螺,或圍繞太陽的行星,故我把它叫做陀螺數魔.此外再由疊羅漢的角度去看,同這個轉軸同方向平行而連續的,或是不連續的所有九條對角線,都必須有九個不同的數字,是另一個限制.

仔細分析這兩個數魔,可以發現它們符合底下的特性:

1. 有九條具有九個不同數字的行,

2. 有九條具有九個不同數字的列,

3. 有九組具有九個不同數字的”三”字組合,

4. 有九組具有九個不同數字的”川”字組合,

5. 有九條平行且有九員不同數字的對角線(一條連續的八條不連續的),

6. 有九條具有相同九員數字且與上一條規則垂直的垂直線(一條連續的八條不連續的),

因此就一共有五十四個九員數字限制.符合這些條件的,就叫陀螺數魔.

伏羲數魔 (Fu Xi Sudoku)比陀螺數魔更增多了兩項限制,既:

7. 兩條連續大對角線的總值相等.

8. 轉軸對角線數字必須按數字順序或大或小排列.這樣就又增多了九條

(一條連續的八條不連續的九員數字)

故伏羲數魔一共符合六十四條有九個數字的限制.這個有超級九員數字限制特性的獎杯恐怕是非伏羲數魔莫屬了.

尤其神奇的是:三川,陀螺,伏羲,及甲子這幾個魔術方塊裡的"三"和"川",就像伏羲八卦裡的”乾卦”.

伏羲數魔本身是不需要任何提示的.在一個完全空白的九九魔術方陣裡,只要知道排列的規則,就能將唯一的八十一個數字組合填滿全部空白.

记得在一九五九年,当我还是初二学生的时候,我的代数启蒙老师刘真真女士,在下课后讲解了几分钟的魔数方块的构造方法.现我模仿她的架构,重新组合,发展出一个新的魔数方块.这个魔数方块,我称它为伏羲数魔 (Fu Xi Sudoku), 以用来纪念伏羲帝五千年前首先发现河图上的神奇数字的组合,及从中演义出来自今仍然全球风行的八卦易经哲理.

伏羲数魔是属于三川魔术方块 (San Chuan Sudoku) 的一种.另外我还做了一道相关 的迷题,叫陀螺数魔(Spinning Top Sudoku) 做为本周的迷面.伏羲数魔与陀螺数魔源于一脉,都出于三川魔术.虽然这道迷题只给了八个已知数,看起来好像是很难,其实却非常容易解的.

伏羲数魔和陀螺数魔主要有两个限制,既:将由西北到东南方向的大对角线视为转轴,每一条向它垂直的线上(总共有十五条垂直线)所有的数字都要相同.但是由叠罗汉的角度去看就一共只有九条连续的或是不连续的垂直而同时还包含九个数的直线,这个奇特的对称性使整个方阵很像一个自转的陀螺,或围绕太阳的行星,故我把它叫做陀螺数魔.此外再由叠罗汉的角度去看,同这个转轴同方向平行而连续的,或是不连续的所有九条对角线,都必须有九个不同的数字,是另一个限制.

仔细分析这两个数魔,可以发现它们符合底下的特性:

1. 有九条具有九个不同数字的行,

2. 有九条具有九个不同数字的列,

3. 有九组具有九个不同数字的”三”字组合,

4. 有九组具有九个不同数字的”川”字组合,

5. 有九条平行且有九员不同数字的对角线(一条连续的八条不连续的),

6. 有九条具有相同九员数字且与上一条规则垂直的垂直线(一条连续的八条不连续的),

因此就一共有五十四个九员数字限制.符合这些条件的,就叫陀螺数魔.

伏羲数魔 (Fu Xi Sudoku)比陀螺数魔更增多了两项限制,既:

7. 两条连续大对角线的总值相等.

8. 转轴对角线数字必须按数字顺序或大或小排列.这样就又增多了九条 (一条连续的八条不连续的九员数字)

故伏羲数魔一共符合六十四条有九个数字的限制.这个有超级九员数字限制特性的奖杯恐怕是非伏羲数魔莫属了.

尤其神奇的是:三川,陀螺,伏羲,及甲子这几个魔术方块里的"三"和"川",就像伏羲八卦里的”干卦”.

伏羲数魔本身是不需要任何提示的.在一个完全空白的九九魔术方阵里,只要知道排列的规则,就能将唯一的八十一个数字组合填满全部空白.

Drjsguo 6-9-2007

Information about San Chuan Sudoku, Jia Zi Sudoku, as well as Dieluohan concept can be found in related web pages on this “Sudoku Club”.  If a reader is confused about Dieluohan concept, just thinking each 9-membered constraint as an equation which consists nine variables with a given know value.

有關三川數字魔術方塊,甲子數字魔術方塊,及疊羅漢觀念請參考本”數獨園”相關的網頁.讀者如果對疊羅漢不能瞭解,可以想像成每一條九員限制就是一個只有九個變數的方程式.

有关三川数字魔术方块,甲子数字魔术方块,及叠罗汉观念请参考本”数独园”相关的网页.读者如果对叠罗汉不能了解,可以想像成每一条九员限制就是一个只有九个变数的方程式.