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Sudoku Myth and Comparison with Marblewits

By Dr. Ifay F. Chang

Recently, a number puzzle fashioning the cross-word puzzle pattern has gained popularity. As an interesting and popular puzzle claiming 'pure logic', Sudoku is deserving some space in our Scrammble Land to be explained and compared with other 'logic' based games. Here we shall describe the Sudoku puzzle and its myth and compare it with the Marblewits game. In two separate articles, we will compare it with Our Noodlewar Games and  Magic Number Games. 

Sudoku Puzzles

Sudoku is a puzzle based on the 'yes or no', 'if and true or false' or simply the 'or' logic. This logic is applied in numerous instances in the real world hence it is an important logic. Mathematically, the 'or' logic is related to probability theory. For example, when you flip a coin, you will get either a head 'or' a tail each with a 50% probability. On the other hand, when you throw a six-sided die with numbers 1, 2, 3,4,5 and 6 on the six faces, you will get any one number with a probability of 1/6. This is the extend of math and logic behind Sudoku. There is no magic nor mystery in it. 

Now, what is a Sudoku Puzzle, in case you have not seen one before? Sudoku is a matrix of 9 by 9 spaces to be filled with 9 distinct numbers, say from 1 to 9 (Can be any 9 different numbers if you wish but 1 to 9 is simpler to remember). There are only three simple rules (conditions) for solving the Sudoku puzzle that is each row , column and 3 by 3 sub-matrix of the 9 by 9 matrix must contain one and only one of the 9 numbers. Some clues are given to make the puzzle either solvable or make the puzzle to have a unique answer rather than multiple answers. We will first use a simple 4 by 4 example to illustrate how the Sudoku puzzles are solved then extend to the general 9 by 9 matrices.

Simple 4 by 4 Matrix 

1

2

3

4

 

1

2

3

4

 

1

2

3

4

         
         

3

4

     

3

4

1

2

         
         

2

       

2

1

         

4

3

         

4

       

4

3

         

2

1

This simple puzzle is to fill four numbers (1 to 4) in the matrix obeying the three basic rules, that is each row, column and 2 by 2 sub-matrix can have only one number from 1 to 4. To solve this puzzle, we simply apply the 'or' logic. Let's first take the first (yellow) sub-matrix, the first cell on the top corner of the sub-matrix may have 1 or 2 or 3 or 4. If there is no clue given, then these four possibilities are equally valid. Hence, you may start anyone. Let's say you place a 1 in the first cell as shown in the first 4 by 4 matrix diagram. Then the remaining 3 cells in the first row can only be 2, 3 or 4 with equal probability. You may pick 2 to fill in the second cell of the first row. Then a 3 in the 3rd cell and 4 in the 4th cell. Similarly you may do that for the first column to put a 3 or 4 in the 2nd cell (2 is not allowed since a 2 is already in the yellow sub-matrix), then a  2 or 4 in the 3rd cell or 4th cell of the first column. Let's say you picked 2 and 4 as shown in the second 4 by 4 matrix. At this point, the 4th cell in the yellow sub-matrix can only be a 4 since 1, 2 and 3 occupied the other 3 quadrants. Let's put in the 4 in the 4th quadrant of the yellow sub-matrix as shown in the second 4 by 4 matrix. Proceeding further, you will have to put 1 or 2 in the two lower cells of the green sub-matrix and 1 or 3 in the right two cells of the orange sub-matrix. The choices are equally good. let's say you picked one possibility as shown in the third 4 by 4 matrix. At this point, only the pink quadrant remains to be filled, however, the choice is already fixed because of the numbers in the other quadrant cells. Let's look at the first upper left pink cell, it can only take a 4 because 1 and 2 can not be placed there due to their presence in the row and 1 and 3 can be placed there due to their presence in the column already. So you must place a 4 in the upper left pink cell as shown in the 4th 4 by 4 matrix. By same logic, only 3 can be placed at the upper right pink cell, only 2 can be placed in the lower left pink cell and only 1 can be placed in the lower right pink cell as shown in the 4th 4 by 4 matrix. Now, it is obvious, superposing the 4th 4 by 4 matrix on the 3rd 4 by 4 matrix gives you a correct answer. 

In this illustration, you can see the correct answer is not unique simply because, you could have started the first row as 2,1,4 and 3 or other allowed sequence. Since there are 4! ( 4 factorial = 4x3x2x1= 24) possibilities just with the first row, there are at least 24 correct answers with different number sequence in the first row. For each instance, there are also multiple possibilities of number sequence in other rows that may fulfill the rules. Hence, this 4 by 4 Sudoku puzzle with no clues (or conditions) would have many correct answers. This makes the puzzle less interesting!.

   How To Create A Sudoku Puzzle

2

 

 

4

 

2

 

 

4

 

2

1

3

4

 

2

1

3

4

  2     4
    2    

 

 

2 1  

 

 

2

1

     

2

1

      2  
      3  

 

    3  

 

 

  3      

1

3         3
1        

1

    2  

1

 

  2   1 3

4

2     1    

It turns out that it is a little more interesting or challenge to create a Sudoku puzzle than to solve one. Since an interesting and solvable Sudoku puzzle must have just adequate clues to make the puzzle solvable with unique answer, a creator has to do more testing to provide just the right clue. For instance, the first 4 by 4 matrix in the above diagram shows 5 numbers in the five cells of the 16 cell matrix. To solve this puzzle, you should start with a cell that have more information or clues. Since there are two numbers in the first row and 4th column, you should start with the empty cells in them first. Let's say, you start with the 2nd cell of the 4th column since it has the most clues (information). This cell can only be 1 or 2 since 3 and 4 are already present in the column. The additional information of  a 2 being at the second row dictates that this cell has to be a 1 (can not be a 2). Then the 4th cell in the 4th column has to be a 2 as shown in the 2nd 4 by 4 matrix diagram. You may now determine the 3rd cell of the first row to be a 3 since the upper right green quadrant is missing only a 3. Then the 2nd cell of the first row must be a 1 as shown in the 3rd 4 by 4 matrix diagram. Now you have 9 out of 16 cells solved. The remaining cells now have sufficient clues to be solved. The 3rd and 4th cell of the 3rd row can only be a 1 or 4, but a 1 is already in the 4th row. Hence the 3rd and 4th cell of the 3rd column must be 1 and 4 in that order. Following that, the 2nd cell in the 4th row must be a 3. The 2nd cell of 2nd row must be 4 or 2 but 2 is already in the 3rd cell of the 2nd row. So the 2nd cell of 2nd row and 3rd row are 4 and 2 respectively. The last two cells, 1st cell of 2nd and 3rd row must 3 and 4 respectively. 

As you can see from the illustration, the Sudoku puzzle can be solved with the same logic step by step only as a matter of time (number of steps needed to follow). There is no myth to it! However, if you would like to create a Sudoku puzzle, you must test it with the same logic until you verify that a correct answer is obtainable. This sometimes can be more tedious than solving a correct puzzle. For example, the 5th 4 by 4 matrix in the above diagram may seem to be a valid Sudoku puzzle but it is actually a false one without a solution!  

   Extend Sudoku Puzzle to 9 by 9 Matrix

2

9

1

6

3

7

4

8

5

 

 

       

5

 

2

 

8

3

5

1

9

4

6

2

7

 

5

4

 

2

   

7

   

4

6

7

2

5

8

3

9

1

 

8

   

6

     

4

3

1

7

6

9

2

3

8

5

4

   

2

     

9

 

6

7

5

4

9

8

7

1

2

6

3

         

6

       

3

2

8

5

4

6

1

7

9

 

3

6

 

7

     

8

 

9

1

2

3

8

5

7

4

6

 

7

3

     

6

   

4

6

5

4

7

1

2

9

3

8

     

4

   

1

 

5

9

7

8

3

4

6

9

5

1

2

   

1

 

8

         

The Sudoku puzzle can be easily extended to 9 by 9 matrix. Again, the creation of Sudoku requires tedious testing so that appropriate clues can be provided to render the puzzle solvable. More or less clues makes a puzzle solvable in a shorter time or longer time. The important thing is to make the puzzle definitely solvable and preferably with one unique answer. The above two diagrams show two Sudoku puzzles. The left one is relatively simple one. (Remember the three rules, each row, column and sub-matrix of 3 by 3 must contain only one complete set of numbers from 1 to 9). To solve this puzzle, one first noticed there are five 1's, four 9's, four 2's, three 3's, 6's and 8's, two 5's, 7's and one 4. So it should be simpler to first place the 4 missing 1's. We can apply the three rules to all the blank space with respect of filling in a 1. First, only the 1st, 3rd, 4th and 8th sub-matrices allow to be filled with a 1 since the other five sub-matrices already have a 1. Then apply the row and column rule, only a 1 can be placed at (1,1), (1,3), (3,1), (3,3), (3, 9), (4,1), and (8,5) spots (the indices represent row number and column number counting from top down and left to right). Out of these 7 spots, the last three are the only one in a sub-matrix, hence, they must be filled with a 1 there. With these 3 1's determined, the only allowable 1 in the first sub-matrix is the (1,3) because the others are excluded by the presence of other determined 1's. These four 1's are shown in red in the above left matrix. Next we examine the 9's. Only five sub-matrices are allowed to be filled with a 9 at (2,5), (4,4), (4,7), (4,9), (6,9), (7,1), (7,7), (7,9), (8,7), (8,9). Applying the row and column rule, the four permissible 9's are (2,5), (4,4), (6,9), ((7,1), and (8,7). The four 9's are filled in green as shown in the above left matrix. Similarly, only five sub-matrices can be added with a 2 based on the sub-matrix rule and they are (1,1), (2,1), (2,2), (2,8), (3,1), (3,2), (3,4), (6,1), (6,2), (9,4), (9,9). By applying the row and column rules, then the permissible 2's are (1,1), (2,8), (3,4), (6,2), and (9,9) as shown in blue. At this point, you can easily spot some obvious placements, for example, the two cells in the 3rd sub-matrix, (1,7) and (2,7) must be 4 and 6 respectively as shown in orange in the diagram. Then in the first row, the 5th and 6th spots must be 3 and 7 respectively. (2,6) must be 4 and (3,5) must be 5. They are shown in orange as well. Following this, then the center sub-matrix must be filled with 7 and 6 at (5,5) and (6,6) respectively, also shown in orange. Next obvious spot is (7,6) which must be 5 as the only number missing in the 6th column. Then in the 5th column the two missing numbers 6 and 8 must be placed at (9,5) and (7,5) as shown in orange color. Now in the 8th sub-matrix, only 4 and 7 are missing, they must be placed at (9,4) and (8,4) to satisfy the row and column rules. Next, the 9th row should be filled since there were only two numbers, 3 and 5 missing which by checking the vertical column must be placed at (9,3) and (9,7) as shown in orange. Next simple target is the 7th column which is missing 7 and 8 at (7,7) and (4,7) for the reason that an 8 is already at (7,5). Next simple target is the 7th sub-matrix which is missing 4 and 5 at the two possible spots (8,2) and (8,3), but, you can not be sure which number is at which spot. You will have to resolve this when more information is obtained. However, this partial information can help you to determine that ((8,8) and (8,9) must be filled with 3 and 8 respectively as shown in orange. In turn, the spot (6,8) must be a 7 by observing the 6th row missing 3, 7 and 8. Since 8 is already in the 6th sub-matrix, and 3 is already in the 8th column. And the two remaining spots on 6th row, (6,1) and (6,3) must be 3 and 8 respectively by checking their corresponding column. Keep searching for easy target, you may find that the two spots (7,8) and (7,9) must be 4 and 6 but not certain which number fits in where. Again, you have to resolve this later with more information. However, this partial information is helpful to resolve the four missing numbers in the 6th sub-matrix where 3,4,5 and 6 are missing. First 3 has to be at (5,9) since it can not be at other 3 spots, (4,8), (4,9) and (5,8). Then by logic, 6 must be at (5, 8) since it can not be at (4,8) and (4,9). Following this determination, you can place 5 and 4 at (4,8) and (4,9) respectively based on the fact that a 5 is already in the 9th column. The next easy target is the (4,2) which must be number 7. At this point, you can also resolve (7,8) and (7,9) to be 4 and 6 because 4 and 6 have been determined to be at (4,9) and (5,8). You have already got 73 out of 81 spots nailed down. The remaining 8 spots can be easily resolved by applying one or two logic steps. Now let's revisit the two spots (8,2) and (8,3), if we place 4 and 5 there in that order, do we see any rule being violated? Yes, there is a problem. If 5 is at (8,3), then 4 and 7 have to be placed at (2,3) and (3,3) in the 3rd column. Since 4 and 7 are already in the 2nd row, hence, you can conclude that 5 must be placed at (8,2) and 4 at (8,3). With this, then you can place 5 and 7 at (2,3) and (3,3)  and consequently 5 and 4 at (5,1) and (5,2) respectively. The remaining 4 spots (2,1), (3,1), (2,2) and (3,2) can then easily be determined to be 8, 4, 3 and 6 by checking the numbers already existing on each row.

The difficulty of a Sudoku puzzle is simply a function of clues given (numbers already given in a matrix). The more clues the easier (take less steps and time to solve) and the less clues the harder. One can not offer too few clues in creating a Sudoku puzzle since most likely the puzzle will have multiple answers which sometimes makes the 'solving process' luck dependent. Hence, one must test a puzzle (trace the logic steps) to make sure the outcome is not too obvious and the process not too tedious (low probability for hitting a right direction). Anyone can create a Sudoku puzzle, the second 9 by 9 matrix in the above diagram is one example for you to solve. All Sudoku puzzles are solved by the same logic process, no myth to it and they do not increase a hell of lot of brain power as commercial claims want you to believe. They are good solitaire games for passing time.          

By the above illustration, one can see that the Sudoku puzzle does induce you to use your brain to apply the same logic rules, hence, it is a brain exercising game. However, we do not consider the game as a Headutainment (Health, Education and Entertainment) game simply because it is not really a significant learning game. The Sudoku puzzles are tedious problems requiring application of the same simple logic rules repetitively. They require a lot of time for human brains to solve them but learning nearly nothing at the end of the game. They may have value in training people to concentrate but do prepare to spend a lot of time in playing Sudoku games without gaining any knowledge. The Sudoku problems are easily solved by a computer since computers are very fast in solving problems using 'if and true or false' or 'or' type of logic especially for problems using repetitive logic steps. Human brains are slower than computer but much more analytical with ability to reason and analyze problems that have not occurred before. I would not recommend Sudoku for young students since its return (knowledge and learning) on investment (time and energy) is not high. Young children are much better off with other learning games. For adults, especially seniors, Sudoku would be like a treadmill for brains, a good exercise game but you must accept its boredom or dreariness. 

Anyone can create a Sudoku puzzle, the second 9 by 9 matrix in the above diagram is one example for you to solve so you may appreciate the points made above.                

Comparison of Sudoku Puzzles and Marblewits

Marblewits games are mathematical logic games. They have roots in the NIM theory (more reference). They are principally two people game, but some games may be played by more than two people. One can create many different game patterns simply by using a game board and a bunch of marbles. The following are four examples of Marblewits. In comparison to Sudoku games, Marblewits offers more varieties so that players will not get bored easily. Marblewits are competitive games between you and your opponent. The games give players opportunity to think about the mathematics behind each game which can lead to combinatorial game theory and analytical theory of other games such as Chess and Wei Chi.  

Like Sudoku, the rules of Marblewits are very simple. The rules for the four games are as follows:

1. Pentagon: Permissible moves are removing any one color or two of green and red marbles. One who removes the last one marble wins.

2. Hexagon: Permissible moves are removing one, two or three Adjacent marbles in a straight line or any number of marbles in a straight line bridging over the Earth planet (Do not remove Earth planet). One who removes the last marble wins.

3. Square: Permissible moves are removing same color marbles, one, two or three continuously linked in a straight line. One who removes the last marble wins.

4. Triangle: Permissible moves are removing same color marbles, one or two adjacent to each other or three in smallest triangle. One who removes the last marble wins.

Unlike Sudoku, Marblewits games are not a solitaire game. The above rules can also be reversed that is one who removes the last marble loses.   

All Icons, Names & Graphics of Marblewits (Trade Mark Intended)

All Rights Including Copyright of  The Marblewits Games Are Reserved 2004

Examples of Marblewits Games  

    

      

Marblewits games can be ordered online.  Use the Scrammble online order form to purchase, the cost is $12.50 excluding tax, shipping and handling. Please include the order item name MBW (Marblewits). The game includes four games, board, marbles, instructions and idea for creating more Marblewits games.  You are welcome to write to the inventor scrammble@scrammble.com  to explore new Marblewits games. For volume order and/or licensing, please write to scrammble@scrammble.com. Special discount is available for teachers and school orders.

To see other games created by Dr. Chang, Click here. Return to Scrammble.com