My first puzzle is not very difficult to solve because I provide far too many digits in order to accommodate four imbedded simplified Chinese characters (China Web-Sudoku, 中华网独). Interestingly, there are not one but 64 solutions for this puzzle (有六十四個解). In the future, I promise the game will be much more up to standard as a real challenge each time. I truly hope this game can provide more fun for the reader.

  More about Dieluohan

  http://mysite.verizon.net/res7seqq/id10.html

  More abut 5-ring Sudoku

  http://mysite.verizon.net/res7seqq/id2.html

  More about 4-ring Sudoku

  http://mysite.verizon.net/res7seqq/id1.html

I may be producing too many new Sudoku variants in recent months.  Readers are probably already having new rules fatigue.  I believe now is the time to pause a little and look back in other interesting directions.

Standard Sudoku has been proven that it must contain a minimum of 17 digits in order to solve it logically.  A good source of 17- digits Sudoku collection can be found at http://people.csse.uwa.edu.au/gordon/sudokumin.php.  Since many new Sudoku variants emerge every week, looking for new minimum digits logic Sudoku must be very exciting and challenging.  Additional restrictions or rules applied to a Standard Sudoku certainly will reduce the number of digits that are required to classify it as logic Sudoku.

Henry’s Rectangle Outline Sudoku is a wonderful logic Sudoku variant with only 15 given digits.  This particular Sudoku has two added rules on top of a Standard Sudoku.  This makes me to believe that the more rules added to a Sudoku, the lesser is the minimum digits that are required to solve it logically.

 Last month, I published two Mudan Sudokus.  A glitch sneaked through my attention while deciding which rules to choose for the final version.  The Mudan Standard Sudoku published was one digit short of logic Sudoku.  I only found out later that it has four answers.  However, the puzzle itself contains 16 digits, low enough to be a candidate for the minimum-digits logic Sudoku quest.  In addition, its rules seem to be complicated enough to find a lower minimum number.  I decided to give it a try.  It did not fail me.  I did it by logic subtraction.  The number came out is “14”.  After successful finding the result, I tried a Mudan Rainbow Sudoku, which I had developed at the same time as the other two were published.  I was very pleased being able to further reduce that number to just “11”.  Is “11” the ultimate minimum number?  I doubt it.  If anyone seen a lower number than this please let me know.

For each individual rules, please follow the links to the rules that listed on this page.  Mudan Rainbow Sudoku has not been published, but the rule is the same as if Mudan Standard Sudoku and Mudan Plum Sudoku put together.  I am not sure if you have already noticed that I have two different patterns assigned to Rainbow and Plum Sudokus in order to avoid the confusion.  (Hint for this minimum logic puzzle: red areas use only digits 1~4). 

有見於讀者們可能每週都要學習魔術方塊新規則,會心生厭煩.故想到, 應該歇息一下, 偶爾找個能提高大家興趣的題目談談.

本週就說說"十七"這個魔術方塊的邏輯門檻."十七"真是個神奇的數字.在傳統數字魔術方塊裡頭, 它是邏輯推演題目裡最低的要求數字.低於它, 就無法應用邏輯推演解題. (底下是一個搜集十七個數邏輯魔術方塊很全的網站: http://people.csse.uwa.edu.au/gordon/sudokumin.php)

新的魔術方塊變種最近如雨後春筍,發展得很快.尤其是鑲有中國字的變種, 很可能更會波濤洶湧.一波接一波地出現.既然新變種增加了許多條件, 十七這個數, 自然就要下滑.雲深去年十一月, 就已經在本刊發表了一道只有十五個數字的長方框數獨題.最近我自己也想嘗試一下, 尋求更少數字的魔術方塊.所以就找了個比較複雜的牡丹花傳統數字魔術方塊來試驗.果然不出所料, 數字又下降一碼, 到了十四. 後來再試一個上個月未推出,但更複雜一些的牡丹彩虹數魔.結果更令人驚奇, 數字居然降低到"十一". 這難道會是極限嗎? 筆者相信更少的數字或許會出現.如果那位讀者發現了新的記錄, 可別忘了通知我.

對於個別問題的規則,請參考本頁的聯接鏈. 雖然牡丹彩虹數魔從未發表過,但它的規則可將牡丹花傳統數魔與牡丹花梅花樁數魔加在一塊就成. 為了分別彩虹數魔及梅花樁數魔,筆者已運用不同的花色以示區別. 希望讀者留意. (本週邏輯題解題題示: 紅色部分請用數字一到四)

有见于读者们可能每周都要学习魔术方块新规则,会心生厌烦.故想到, 应该歇息一下, 偶尔找个能提高大家兴趣的题目谈谈.

本周就说说"十七"这个魔术方块的逻辑门槛."十七"真是个神奇的数字.在传统数字魔术方块里头, 它是逻辑推演题目里最低的要求数字.低于它, 就无法应用逻辑推演解题. (底下是一个搜集十七个数逻辑魔术方块很全的网站: http://people.csse.uwa.edu.au/gordon/sudokumin.php)

新的魔术方块变种最近如雨后春笋,发展得很快.尤其是镶有中国字的变种, 很可能更会波涛汹涌, 一波接一波地出现.既然新变种增加了许多条件, 十七这个数, 自然就要下滑.云深去年十一月, 就已经在本刊发表了一道只有十五个数字的长方框数独题.最近我自己也想尝试 一下, 寻求更少数字的魔术方块.所以就找了个比较复杂的牡丹花传统数字魔术方块来试验.果然不出所料, 数字又下降一码, 到了十四. 后来再试一个上个月未推出,但更复杂一些的牡丹彩虹数魔.结果更令人惊奇, 居然数字降低到"十一".这难道会是极限吗? 笔者相信更少的数字或许会出现.如果那位读者发现了新的记录, 可别忘了通知我.

 对于个别问题的规则,请参考本页的联接链. 虽然牡丹彩虹数魔从未发表过,但它的规则可将牡丹花传统数魔与牡丹花梅花桩数魔加在一块就成. 为了分别彩虹数魔及梅花桩数魔,笔者已运用不同的花色以示区别. 希望读者留意.  (本周逻辑题解题题示: 红色部分请用数字一到四)

I would like to dedicate these two beautiful and very special Mudans to my parents, General and Mrs. Teng Long Guo (郭騰龍將軍及夫人).  Photo on the right was taken in 1954 at Xindian, Taipei (臺北tai bei, 新店xin dian).

Drjsguo 3-3-07

Drjsguo 3-9-2007

After published San and Chuan Sudoku last week, I managed to make a very complicated highly restrained Sudoku.  It is a wonderful feeling of discovering this sixty 9-digit restrained Sudoku.  This Sudoku contains the following restrictions:

    1.  Every row must uses 9 different digits. 

    2.  Every column must uses 9 different digits.

    3.  Every 3X3 box must uses 9 different digits.

    4.  Every Plum position must uses 9 different digits.

    5.  Every San position must uses 9 different digits.

    6.  Every Chuan position must uses 9 different digits.

    7.  Every diagonal starting from the upper corners of a 3X3 box must uses 9 different digits.

The restriction rules from 1 to 6, each has nine different arrangements.  The diagonal has six arrangements.  Total is sixty, thus I give it the name Jia Zi (甲子數魔), which means sixty years in Chinese.  I was thinking to name it Fu Xi (伏羲) Sudoku or Magic (神奇) Sudoku for such a complicated sixty degreed 9-digits rules.  But I think Jia Zi is more appropriate, because I do not know if this highly restrained record of "sixty" has already been broken or will be broken in the future.

Beside the familiar two primary nine-digit straight-line diagonals, there are four additional secondary nine-digit diagonals, which are not easily recognized.  These secondary diagonal lines are discontinued at one of the two vertical edges of Sudoku, but continued at the other end of the vertical edge.  If we use Dieluohan (疊羅漢) expansion technique it is easy to see them as genuine nine-digit straight diagonal lines. 

My first logic puzzle of this kind contains nine known digits.  I was surprised it did not go down further.  However, consider it only has sixty restrictions; the Sudoku still got plenty of freedoms to fill the numbers.  I have already found numerous combinations of Jia Zi Sudokus. 

繼上個星期發表三川數魔以來,很興奮又發現了一個具有六十個九數限制的數魔.

這個數魔具有下列七個特點:

一. 每一行必須用全部九個數.

二. 每一列必須用全部九個數.

三. 每一個三乘三九宮必須有全部九個數.

四. 每一組梅花樁位置必須有全部九個數.

五. 每一組三字的位置必須有全部九個數.

六. 每一組川字的位置必須有全部九個數.

七. 每一條由最上層三乘三九宮上起畫的九數對角線必須有全部九個數.

頭六個限制都各有九種.第七個限制只有六種.其中除我們常見的九數大對角線外,還有四個.這四個是不連續的大對角線.如果您用疊羅漢定律將它向四方擴大,就能看到它們的直線性.

因為一共有六十條九數限制,故我把它起名叫”甲子數魔”.

本週的邏輯題一共只給九個已知數.原本希望可以給得更少.其實八十一格中只有六十個限制,理當還有無限變化的.

继上个星期发表三川数魔以来,很兴奋又发现了一个具有六十个九数限制的数魔.

这个数魔具有下列七个特点:

一. 每一行必须用全部九个数.

二. 每一列必须用全部九个数.

三. 每一个三乘三九宫必须有全部九个数.

四. 每一组梅花桩位置必须有全部九个数.

五. 每一组三字的位置必须有全部九个数.

六. 每一组川字的位置必须有全部九个数.

七. 每一条由最上层三乘三九宫上起画的九数对角线必须有全部九个数.

头六个限制都各有九种.第七个限制只有六种.其中除我们常见的九数大对角线外,还有四个.这四个是不连续的大对角线.如果您用叠罗汉定律将它向四方扩大,就能看到它们的直线性.

因为一共有六十条九数限制,故我把它起名叫”甲子数魔”.

本周的逻辑题一共只给九个已知数.原本希望可以给得更少.其实八十一格中只有六十个限制,理当还有无限变化的.

Drjsguo 5-20-2007   

Please refer to other web pages of “The Sudoku Club” about Plum position, San position, Chuan position, and Dieluohan theory.

有關梅花樁位置,三字的位置,川字的位置,及疊羅漢定律請參考本”數獨園”網站相關的資料.

有关梅花桩位置,三字的位置,川字的位置,及叠罗汉定律请参考本”数独园”网站相关的资料.

This week’s Sudoku is not an ordinary Rainbow Sudoku.  When you solved the puzzle you will find that it has a total of eighteen ordinary or pseudo diagonals, all have the same sum of 45 but with many repeated digits.  As one member of the Rainbow Sudoku family, each row, column, box group, and Plum group are all having nine different digits.  There are additional nine-digit sets with sum also equal to 45 but with few repeated digits: three from San group and three from Chuan group.  So the total number of constraints for this Sudoku is also 60, same as the Jia Zi Sudoku.  This Sudoku is very special and deserves a special name, so I name it “Eighteen Luohans (十八羅漢, shi ba luo han) Sudoku”.

Luohan (羅漢) is Buddhist arahant.  It was originated from India.  Luohan is named for the perfected person who has achieved nirvana and freed himself from the bonds of desire.  He is the guardian of Heavenly law and is exempted from the cycle of rebirth.  When Buddhist religion flourished in China in Tang Dynasty, Luohans became the major subject of art in painting, clay work, and wood sculpture.  Originally there were sixteen Luohans, the number later increased to eighteen, and finally became 500 (五百羅漢, wu bai luo han).

 In western United States, a Buddhist temple named Hsi Lai Temple (西來寺, xi lai si) that is located in Hacienda Heights (East of metropolitan Los Angeles), California.  You may be able to find Luohans there.  In many museums all over the world you can also find Luohan paintings or art works in exhibition.  Photo on top is a picture of Luohan Hall in Shaolin Temple (嵩山少林寺, song shan shao lin si) in Henan.  There is also a giant wall painting of 500 Luohans from Ming Dynasty in Shaolin Temple.  Shaolin Temple was granted “The Greatest Temple on Earth (天下第一名刹)” by Emperor Tang Tai Zong (唐太宗李世民, li shi min) in Tang Dynasty.

This week, I shall publish two research papers about Magic Square and Sudoku here at “The Sudoku Club” website.  The papers are titled “The Magic Properties of Holy Turtle (神龜, shen gui) Magic Squares” and “San (三), Chuan (川), and Super Constrained Sudokus”.  I hope you will enjoy the new findings as much as I do.

本週發表的十八羅漢魔方,是屬於彩虹魔方 (Rainbow Sudoku) 的一個特例.它具有彩虹魔方全部的特性,既每行,每列,每個九宮,每個梅花樁都必需用九個不同數字.

另外,它全部的十八條九員大對角線(連續的及不連續的)之和也都等於45.此外它還具有三個”三組”及三個”川組”它們的九員之和也是45.不過這24組中的九員數字都不是單一的,是會重復出現的.這個十八條大對角線很特別,同時因它也具有60個九員限制,需要一個特別的名稱.故給它起名叫”十八羅漢魔方”.

另外我在本網站將發表兩篇有關九九魔方的新論文.題目是:”洛書神龜 (Holy Turtle) 魔方的特性 (The Magic Properties of Holy Turtle Magic Squares)” 及”三,川,及超多限制的數魔 (San, Chuan, and Super Constrained Sudokus)”.希望讀者會喜歡論文裡的一些新發現.

本周发表的十八罗汉魔方,是属于彩虹魔方 (Rainbow Sudoku) 的一个特例.它具有彩虹魔方全部的特性,既每行,每列,每个九宫,每个梅花桩都必需用九个不同数字.

另外,它全部的十八条九员大对角线(连续的及不连续的)之和也都等于45.此外它还具有三个”三组”及三个”川组”它们的九员之和也是45.不过这24组中的九员数字都不是单一的,是会重复出现的.这个十八条大对角线很特别,同时因它也具有60个九员限制,需要一个特别的名称.故给它起名叫”十八罗汉魔方”.

另外我在本网站将发表两篇有关九九魔方的新论文.题目是:”洛书神龟 (Holy Turtle) 魔方的特性 (The Magic Properties of Holy Turtle Magic Squares)”及”三,川,及超多限制的数魔 (San, Chuan, and Super Constrained Sudokus)”.希望读者会喜欢论文里的一些新发现.

Drjsguo 6-24-07