​If you want to be a savvy Sudoku solver, this technique is a must to grasp: XY-Wing.

Simply speaking, when one square with candidates {X, Y} is in the same row, column, or box as two squares with candidate numbers {X, Z}, and {Y, Z}, candidate number Z can be eliminated from the squares in the same row, column, or box as these two squares.

​Candidate Numbers (or Candidate Values) of a blank/empty square is a list of ‘possible values’ or candidates for this blank/empty cell. Below is an example of using XY-Wing to refine the candidate values of R3C6. Empty Squares are marked with candidate values.

For R5C5, 

​Either way, either R3C9 or R1C4 will be number 1.
​Therefore, R3C6 who is in the same row as R3C9 and at the same box as R1C4 cannot be number 1. We can safely remove number 1 from R3C6‘s candidate values. These above two possibilities rule out the possibility that number 1 could be in R3C6. We can safely remove number 1 from R3C6.

How to find them?

Try this hard Sudoku on 03/25/2018 to see whether you can apply XY-Wing as well as other Sudoku solving strategies to solve it?

​Sudoku on 03/25/2018 on www.createclassicsudoku.com

Solve Hard Sudoku on 03/25/2018 in a step-by-step style

数独是是一种运用纸、笔进行演算的逻辑游戏。玩家需要根据9×9盘面上的已知数字，推理出所有剩余空格的数字，并满足每一行、每一列、每一个粗线宫（3*3）内的数字均含1-9，不重复。

数独游戏有一些有趣事实：

1.  制作数独游戏不需要您一定是专家。任何一个具有基本逻辑推理能力的人都可以在几分钟之内制作一个数独游戏。 

2.  对的，要使得数独游戏只有一个解，给定的已知数字至少要有17个。

3.  一个空的数独游戏一共有 6670903752021072936960 解。 但是，本质上不同的解仅有5,472,730,538. 毫无疑问，您一辈子也玩不完所有的数独游戏。

4. 数独是一种逻辑游戏，绝对不涉及数学或语言技能。

5. 实际上，数独不是日本的游戏。这是美国人发明的。霍华德加斯在1979年称为填数字（Number Place），但在日本发行商尼科利（Nikoli公司）掌握之前于1989年去世。直到2004年韦恩古尔德(Wayne Gould)说服伦敦泰晤士报(Times in London) 出版才真正火起来。

6. 数独在2005年成为世界热点时，据估计这是自20世纪80年代自魔方创立以来最大热点。

7. 自2006年3月以来，每年都有一次全球数独锦标赛。第一届世界数独锦标赛在意大利卢卡(Lucca, Italy) 举行。

8. 根据一些研究，定期玩数独可以带来好处，比如提高注意力和注意力，预防或缓解抑郁症，甚至可能预防痴呆症和阿尔茨海默病。

9. 数独对任何人和任何年龄都有好处，有助于培养智力并保持良好状态。

10. 数独被认为是非常容易上瘾的，但由于没有任何有害的副作用（实际上是一系列极好的副作用），所以不用害怕上瘾！

Dr. Yaling Zheng
From https://www.createclassicsudoku.com/What_are_interesting_facts_about_Sudoku.jsp

Sudoku is a number game in which missing numbers are to be filled into a 9 by 9 grid of squares
which are subdivided into 3 by 3 boxes so that every row, every column, and every box contains
the numbers 1 through 9. 

Here are some interesting facts about Sudoku:

1. You don’t need to be an expert to make a Sudoku puzzle. Anyone with basic logical reasoning
can make a Sudoku puzzle within minutes.
2. Yes, there is a minimum number of clues to be given for the Sudoku puzzle has one solution.
The least number of clues of a given Sudoku with a unique solution is 17.
3. There are 6670903752021072936960 Sudoku grids. However, the essentially different Sudoku
grids are only 5,472,730,538. Needlessly to say, you need a handful of lifetimes to solve all of
them.
4. Sudoku is a logic game and involves absolutely no math or language skills.
5. Actually, Sudoku isn’t a Japanese game it all. It’s American invented. Howard Garns created
it as Number Place in 1979 but died in 1989 before Japanese publisher Nikoli got a hold of it.
The game didn’t really take off until 2004 though when Wayne Gould convinced The Times in
London to publish it.
6. When Sudoku became a world hit in 2005, it is estimated that it is the biggest phenomenon
since the Rubik’s Cube in the 1980s.
7. There is a worldwide Sudoku Championship every year since Mar 2006. The first World
Sudoku Championship was held in Lucca, Italy.
8. Playing Sudoku regularly can have benefits, like boosting your concentration and focus,
preventing or easing depression and possibly even preventing dementia and Alzheimer’s disease,
according to some studies.
9. Sudoku is good for anyone and any age and helps develop mental abilities as well as keeps
them in good condition.
10. Sudoku is considered highly addictive, but since there aren’t any harmful side effects (and in
fact a list of great side effects), go right ahead and get addicted!

     郑亚玲博士

        锻炼自己或者孩子的观察力，耐心，还有逻辑推理能力？推荐给你一个智力挑战  ― 设计一个数独游戏。
数独是一个数字逻辑推理游戏（见该文末尾的今日数独）。给定一个9行9列9个宫的大方格。一共是81个小方格。一部分小方格已经填好数字。 游戏的目标是填充所有空方格使得每行，每列，每个宫包含9个不同数字 {1，2，3，4，5，6，7，8，9}。一个经典数独有并且只有一个解。请参见今日数独。 请花上一点时间思考：如何能制作这样一个游戏？ 如何用最快的速度设计一个这样的游戏？

        比较直接的方法是   
       (1)  画好（打印）一个九行九列九个宫的空的大方格
​     (2)  选中一部分小方格（这些小方格将预先填写数字）
​       (3) 小心地往这些选中的方格中填写数字（注意不要违反每行，每列，每个宫一个数字只出现一次的规则）；如果违反规则，重新填写。
​       (4) 检查这个设计的数独游戏是不是只有唯一解。如果不是，重复2或者3。

       缺点:  这种方法虽然直接但是成功制作的概率非常低（比赢彩票的概率还要低）。连第3步都可能重复多次。当你尝试了几次之后，你可能就想放弃。
​
       那么有没有一种更有效率的方法呢？有的。这个设计理念就是“和玩数独完全相反的方向”。我们先填满所有方格，然后小心翼翼地去掉一些数字。

      有好多个捷径可以填满所有方格。其中一种方法是填写9个不同数字 (group filling) 到第一行 (比如369287145)。然后，循环移位这九个数字(circular shift)填写到下一行(比如287145369)；重复循环移位填写第3行(比如145369287)；第4行(比如692871453) ； 第5行(比如871453692) ；第6行(比如453692871) ；第7行(比如928714536) ；第8行(比如714536928) ；第9行(比如536928714)。
     
       小心翼翼地去掉数字的时候，检查是不是安全去除（safe removal）― 如果我们去掉这个数字，能够根据留下的数字推理出它吗？如果是，就是安全去除。如果一个数字可以安全地被除掉，我们就可以去掉它。重复这个安全去除步骤直到所有小方格数字都被检查过。详细的细节请参考Create Classic Sudoku – Make Your Own in Minutes 
       
        用这种方法，你可以几分钟之内制作一个非常容易的数独游戏。是不是很神奇？下面是一个用这种方法制作的一个中等难度的数独。更多数独？请访问www.createclassicsudoku.com.

今日数独  （难度：中等）

郑亚玲博士

      无论您是在旅行还是正在医院康复，数独都是一款完美的老少皆宜的益智游戏！它有助于培养您的逻辑推理能力，耐心，专注解决问题的能力，和自信心。
​
      下面是一个强迫链的例子: 注意到空格R4C4 (位于第四行第四列的空格),然后我们尝试让R4C4是数字3或者数字4（数字3和4是R4C4的可能数字）。下面你会看到，无论R4C4是数字3还是数字4， R1C3都必须是数字4。

​      如果R4C4是数字4, R1C4 必须是数字8 (step 1), 进一步推理R1C3必须是数字4。

       如果R4C4是数字3， R4C7必须是数字4 (step 1)， 进一步推理R5C9必须是数字 3(step 2)， 进一步推理R3C9必须是数字4 (step 3)， 进一步推理R3C3必须是数字8，最后推理出R1C3必须是数字4。

       因此，无论R4C4是哪个备选数字, R1C3必须是数字4。这个例子成功诠释了如何利用强迫链来确定某个空格必须是某个数字。强迫链是解高难度数独的一个方法。这个方法需要你有足够的耐心来找到强迫链的起始空格。 

今日数独 (高难度）

       关于作者：亚玲是制作经典数独 Create Classic Sudoku(在亚马逊热卖)的作者和万维网www.createclassicsudoku.com 网站创始人。她同时也是Cleveland/Akron地区的房产经纪人。她的电话是216-245-3258 短信或电话。微信号是 yalingzheng2013。

Dr. Yaling Zheng

Whether you are traveling, taking a break, or recovering in hospitals, solving a number puzzle game (Sudoku Puzzle)  would be a perfect brain workout for you! 

When you solve a hard Sudoku puzzle, forcing chain method can help you determine exactly what number a certain empty square must hold.

Forcing chains work in the following way: Start on the square R4C4 (the square in the fourth row and the second column), and fill in one of the two candidates, 3 or 4. The following I am going to show no matter R4C4 is 4 or 6, R1C3 will always be number 4. This is exactly a great example of forcing chain. 
​
If R4C4 is number 4, R1C4 has to be number 8 (step 1), and then R1C3 has to be number 4.

If R4C4 Iis number 3, R4C7 has to be number 4 (step 1), and then R5C9 has to be number 3 (step 2), and then R3C9 has to be number 4 (step 3), and then R3C3 has to be number 8, and then R1C3 has to be number 4.

Therefore, no matter what number R4C4 is, R1C3 will be number 4. This is a great example of using forcing chain to determine a certain number for an empty square.  Forcing chain is an advanced method for expert Sudoku solvers. It takes time and patience to find forcing chain patterns. And I only recommend you use this method if all other simpler methods do not work. 

Today's Sudoku (Difficulty Level: Hard)

郑亚玲博士
​
       无论您是在旅行还是正在医院康复，数独都是一款完美的老少皆宜的益智游戏！它有助于培养您的逻辑推理能力，耐心，专注解决问题的能力，和自信心。设计数独的其中一个环节是填满空数独。参见图一。​

图1: 填充空数独。

​如何填写数字到每个空格，使得每行，每列，每个三乘三的宫格包含数字1到9呢？两个诀窍。

分组填写：一次填写九个不同数字到一行，或者一列，或者一个宫格。这样你就只有九个小任务，而不是八十一个小任务（每次填写一个空格）。

循环位移 Circular shift – 每次将前一行（或列或宫）的数字做一个循环位移填写到当前行（或列或宫）。

图2：九个步骤填充每个空格.

第一步： 填写九个不同数字692453781到第一行。

第二步： 将692453781循环右移三个位置，即 453781692，781稍做变换排列为817，692稍做变换排列为269 (注意：这两个变换都不是必须的)，即 453817269填写到第二行。

第三步： 将453817269循环右移三个位置，即 817269453，817稍做变换排列为781 (注意：这个稍做变换不是必须的)，即 781269453填写到第三行。
​
同样的道理填写第四，第五，第六，第七，第八，第九行，就成功填写了所有的空格了。
想要知道更多设计数独的诀窍？请参考 CREATE CLASSIC SUDOKU一书。在AMZON有售。

今日中等数独

​关于作者：亚玲是制作经典数独 Create Classic Sudoku(在亚马逊热卖)的作者和createclassicsudoku.com 网站创始人。她同时也是Cleveland/Akron地区的房产经纪人。她的电话是216-245-3258。微信号是 yalingzheng2013。

Dr. Yaling Zheng

Whether you are traveling, taking a break, or recovering in hospitals, designing a number puzzle game would be a perfect brain workout for you! 

​One of the steps of making a Sudoku is to fill every square of an empty grid. Please see Fig 1.

Fig 1: Task to fill every empty square.

Did it take you long to achieve this task?

If you can quickly fill every empty square satisfying that every row, column, and box has nine different numbers 1 to 9, please drop me an email telling me how you did it at create.classic.sudoku@gmail.com; if not, would you like to know a shortcut to achieve it?

The shortcut uses two techniques: Group Filling and Circular Shift.
Group filling – instead of fill one square a time, you fill the one row (or column or box). You fill the row (or column or box) with nine different numbers 1 to 9.
Circular shift – every time you fill the row (or column or box) by a circular shift of the sequence in the previous row (or column or box).
​
With group filling, you fill the grid much faster. With the circular shift, you tactfully avoid conflict of the same number in a row or column or box.
​
How did we achieve the result in Figure 1? Nine steps. Please see Fig 2.
​

Fig 2: Nine Steps to fill a Grid.

Step 1:  Fill nine different numbers into the first row. Let us fill the first row with the sequence 692453781.

Step 2:  Perform a circular shift the sequence in the first row by THREE slots. We can fill it with 453781692. We do a permutation of 781 in the second box to 817. And a permutation of 692 to 269 in the third box.  In summary, we fill the second row with 453817269.

Step 3: Perform a circular shift the sequence in the second row by THREE slots. We can fill it with 817269453. We do a permutation of 817 in the first box to 781.  In summary, we fill the third row with 781269453.

Step 4: Perform a circular shift of the sequence in the third row by SIX slots, which is 453781269. But 453 in the first box has a conflict, so we do a permutation to 534. We do a permutation of 269 in the sixth box to 692. In summary, we fill the fourth row with 53478692.

Step 5:  Perform a circular shift of the sequence in the fourth row by THREE slots, which is 7816923534. But 453 in the first box has a conflict, so we do a permutation to 345. We do a permutation of 269 in the sixth box to 692. In summary, we fill the fifth row with 178692345.

Step 6: Perform a circular shift of the sequence in the fifth row by THREE slots, which is 69234 5178. But 692 in the fourth box has a conflict, so we do a permutation to 269. We do a permutation of 345 in the fifth box to 534, and a permutation of 178 in the sixth box to 817. In summary, we fill the sixth row with 269534817.

Step 7: Perform a circular shift of the sequence in the sixth row by THREE slots, which is 534817269. But 5 3 4 in the seventh box has a conflict, so we do a permutation to 345. We do a permutation of 9 6 2 in the eighth box to 926, and a permutation of 269 to 926. In summary, we fill the seventh row with 345178926.

Step 8: Perform a circular shift of the sequence in the seventh row by THREE slots, which is 178926345. But 178 in the first box has a conflict, so we do a permutation to 817. We do a permutation of 345 in the ninth box to 534. In summary, we fill the eighth row with 817926534.

Step 9: Perform a circular shift of the sequence in the eighth row by THREE slots, which is 926 534817. But 534 in the eighth box has a conflict, so we do a permutation to 345. We do a permutation of 817 in the ninth box to 178. In summary, we fill the ninth row with 926345178.

Want to know more about making a Sudoku? Please refer my book CREATE CLASSIC SUDOKU selling on Amazon.com. Here is 

Today’s Medium Sudoku

郑亚玲 博士

       无论您是在旅行还是正在医院康复，数独都是一款完美的老少皆宜的益智游戏！它有助于培养您的逻辑推理能力，耐心，专注解决问题的能力，和自信心。有朋友告诉我，制作数独太复杂了是不可能的。但其实只要你制作数独的一个基本理念―“和玩数独完全相反的方向”，制作数独  其实很简单。怎么个简单法呢？请看下面的一个小例子你就知道了。

       对于不熟悉数独游戏的朋友，什么是数独呢？数独是一种逻辑性的数字填充游戏，玩家须以数字填进每一格，而每行、每列和每个宫（即3x3的大格）有齐1至9所有数字。游戏设计者会提供一部分的数字，使谜题只有一个答案。

        一个经典数独要求这个数独有并且只有一个答案。并且，位于第x 行和第y列的方格（我们称为RxCy）和位于第10-x 行和第10-y列的方格要么同时被填充要么同时没有被填充。(1 ≤ x, y ≤ 9) 例如 R8C1和R2C9要么同时被填充要么同时没有被填充。

​       P1 是一个经典数独。解一下数独，是不是只有一个解(答案)？是不是位于位于第x 行和第y列的方格和位于第10-x 行和第10-y列的方格要么同时被填充要么同时没有被填充 (1≤ x, y≤ 9) ？     

经典数独P1​​

​思考一下:  如果我们去掉R8C1和R2C9的数字, 新的数独P2还是一个经典数独吗？答案是肯定的。为什么呢？

首先，我们去掉这两个方格的数字后，这个数独依然满足这个对称的要求：位于第x 行和第y列的方格和位于第10-x 行和第10-y列的方格要么同时被填充要么同时没有被填充 (1 ≤ x, y ≤ 9 ).
其次，如果我们去掉 R8C1 和R2C9的数字，这个新的数独还是只有一个解（答案）。

数独P2​

       R8C1 必须是 9。因为在第一列 ：数字9 不能在R1C1因为第一行已经有数字9；数字9不能在第二行因为第二行已经有数字9；数字9不能在第五行因为它所在的3x3宫格已经有数字9。根据排除法，第一列的空格中，只有R8C1可以是数字9。

       R2C9 必须是数字2。因为在第二行：数字2不能在R2C1或者R2C3因为第一宫已经有数字2了；数字2不能在R2C4因为第4列已经有数字2了。根据排除法，在第二行的空格中，只有R2C9可以是数字2。
​
        根据推理，我们可以将数字9和数字2填到R8C1和R2C9。这个填写数字后的数独，正好是经典数独P1。因为P1有且只有一个解，所以我们可以说数独 P2有且只有一个解。

        思考：对于数独P2, 如果去掉 R4C6 和R6C4 的数字，新的数独P3还是一个经典数独吗？
​

​        所有createclassicsudoku.com 的数独都是根据这个理念―“和玩数独完全相反的方向”设计的。来试一下今日数独？

今日数独 （难度：中等）

Dr. Yaling Zheng

Whether you are traveling, taking a break, or recovering in hospitals, designing a number puzzle game would be a perfect brain workout for you!  I have met friends who think making a Sudoku puzzle is almost an impossible mission.  However, if you know the basic principle to make a Sudoku puzzle – REVERSE WAY OF SOLVING A SUDOKU PUZZLE, making a Sudoku will become easy for you!  In this article, I will talk about one trick to make a classic Sudoku. 

Let us first take a look at the definition of a Sudoku. Sudoku is a puzzle in which missing numbers are to be filled into a 9 by 9 grid of squares which are subdivided into 3 by 3 boxes so that every row, every column, and every box contains the numbers 1 through 9. 

A "Classic" Sudoku is a symmetrical Sudoku that has one and ONLY one solution.  A symmetrical Sudoku is a Sudoku in which any two symmetrical squares (around the center square) are either both filled with numbers or both empty.  In the following we use RxCy to represent the square at the xth row and the yth column. R8C1 and R2C9 are symmetrical squares around the center square R5C5.   Here is an example of classic Sudoku. We call it Sudoku Puzzle P1.  

Sudoku Puzzle P1

​This Sudoku is a symmetrical Sudoku and it has one and only one solution (you can check that out by solving it).
Question:  If we remove numbers from R8C1 and R2C9, is it still a classic Sudoku? 
The answer is “Yes”. Why?
First, if we remove numbers from R8C1 and R2C9, it is still a symmetrical Sudoku.
Secondly, if we remove numbers from R8C1 and R2C9, the new Sudoku still has one and only one solution.  Why?

Sudoku puzzle P2​

R8C1 can be immediately inferred to be number 9 because in the first column
(1) Number 9 cannot be in the first row because the first row contains number 9 already;
(2) Number 9 cannot be in the second row because the second row contains number 9 already;
(3) Number 9 cannot be in fifth row because the fourth box contains number 9 already;
(4) number 9 can only be in the eighth row because it cannot be in the other empty squares in the first column.

R2C9 can be immediately inferred to be number 2 because in the second row
(1) Number 2 cannot be in the first or third column because the first box contains number 2 already;
(2) Number 2 cannot be in the fourth column because the fourth column contains number 2 already;
(3) number 2 can only be in the ninth column because it cannot be in the other empty squares in the second row.  

If we fill numbers 9 and 2 into R8C1 and R2C9 separately in Puzzle P2, it becomes puzzle P1.  Since puzzle P1 has one and only one solution, we can declare that puzzle P2 has one and only one solution too.

Think:  Remove numbers in R4C6 and R6C4 in puzzle P2, name the new puzzle P3. Is the puzzle P3 a classic Sudoku?

​All Sudoku puzzles made in www.createclassicsudoku.com are made using this basic principle – REVERSE WAY of solving a Sudoku puzzle.  How about try solving today’s medium Sudoku? 

Today’s Medium Sudoku