SUDOKU #1  
 

I've been doing Sudoku for a while. There are a lot of people like me who have figured out some tricks or techniques that help them complete the puzzles. This page is not really for them. I met a woman recently who was just starting to try to do Sudoku. This page is for people like her. I'm not saying that I know every single trick, but I know enough to complete most puzzles.
 
The basic rules of Sudoku are that there is only one of each number in each row, column, and 9 square grid. There isn't any mathematics or arithmetic. If you know how to count from 1 to 9, then you can do these puzzles. All of the techniques are based on this rule.
 
The first technique that I discovered was a combining of the rule that all rows, columns, and boxes have one of each number. I learned that the each set of 3 rows or columns are linked. This is what I mean. Here is a sample Sudoku puzzle.
 
      9     5     6    
        5     6   7    
    4     7     1 3    
    9       1     8 3  
          8   3        
    7 8     4       2  
      4 1     9     7  
      7   1     3      
      3     7     5    
 
This is a medium difficulty puzzle, and it will allow me to show some examples of a few of the tricks that I've uncovered. I've added the letters below to the rows and columns in order to talk about each of the 81 squares. As a convention, I will talk about squares and boxes. There are 81 squares, but there are 9 boxes of 9 squares.
 
  J K L M N O P Q R  
  A   9     5     6    
  B     5     6   7    
  C 4     7     1 3    
  D 9       1     8 3  
  E       8   3        
  F 7 8     4       2  
  G   4 1     9     7  
  H   7   1     3      
  I   3     7     5    
 
A ramification of the rule that there is only one of each number in each 9-square grid is that each set of three rows or columns are connected. Rows A, B, and C are connected. For example, there can only be one 9 in the upper left set of squares, but the same is true for the top center set and the upper right set of squares. That makes three 9s. A 9 is in row A at square AK. The other two sets of squares (AM-CO and AP-CR) would have one 9 each, and one would have to be in row B while the other would have to be in row C. The same can be said for columns J-K. There is a 9 in column K at square AK, and the other is in column J at square DJ. That means that there is a 9 in either HL or IL.
 
Take for example the number 1s in rows G-I. There is a 1 in GL, and another one in HM. So the 1 in row I needs to be in either square IP or IR. But there is a 1 in square CP. So the 1 in row I has to be in IR. Now let's look at columns M-O. There are two 1s: one in column M at HM, and the other in column N at DN. The 1 in column O has to be in square AO, because there is a 1 at square CP.
 
  J K L M N O P Q R  
  A   9     5 1   6    
  B     5     6   7    
  C 4     7     1 3    
  D 9       1     8 3  
  E       8   3        
  F 7 8     4       2  
  G   4 1     9     7  
  H   7   1     3      
  I   3     7     5 1  
 
I like to compare this technique to the concept in chess of rows or columns of influence. One example of the other side of this is the 1 in box DP-FR. Look at the 1 at GL. It influences the whole of column L. That means that squares DL-FL cannot be a 1. But the 1 in DN also means that DK and DL cannot be 1. That means that the 1 in box DJ-FL has to be either in EJ or EK. So the 1 in box DJ-FL makes it so that there can be no other 1 in row E. Now look at columns P and R. They already have their 1s. The 1 in column Q has to be in square FQ, since it cannot be in row E.
 
  J K L M N O P Q R  
  A   9     5 1   6    
  B     5     6   7    
  C 4     7     1 3    
  D 9       1     8 3  
  E       8   3        
  F 7 8     4     1 2  
  G   4 1     9     7  
  H   7   1     3      
  I   3     7     5 1  
 
That's basically all we can do on the 1s. There are two left: one in BJ or BK, and the other in either EJ or EK.
 
Now let's look at the 7s. There is one 7 in column J and one in column K, but there are 7s in rows B and C. So, because of our rule of "rows of columns of influence", there has to be a 7 in square AL. There are 7s in columns M and N, and there is a 7 in row F. Therefore, there has to be a 7 in square DO. There are 7s in column Q and R, but there are 7s in rows D (the new one at DO) and F. So there has to be a 7 in square EP. That finishes off the 7s.
 
  J K L M N O P Q R  
  A   9 7   5 1   6    
  B     5     6   7    
  C 4     7     1 3    
  D 9       1 7   8 3  
  E       8   3 7      
  F 7 8     4     1 2  
  G   4 1     9     7  
  H   7   1     3      
  I   3     7     5 1  
 
So we're making progress, and the more squares you fill in, the less squares are left for the other numbers. We can place a 3 at square FL, because there is a 3 in row D and row E. And we can place a 5 in square CR, because there is a 5 in rows A and B.
 
  J K L M N O P Q R  
  A   9 7   5 1   6    
  B     5     6   7    
  C 4     7     1 3 5  
  D 9       1 7   8 3  
  E       8   3 7      
  F 7 8 3   4     1 2  
  G   4 1     9     7  
  H   7   1     3      
  I   3     7     5 1  
 
Here's another technique that uses a process of elimination. There can't be a 5 in square ER or EQ, and there can't be one in EN. So the 5 in row E has to be in box DJ-FL. It can't be in square EL, but it can't be in EJ either, because the 5 in box GJ-IL has to be in column J. So the 5 in row E has to be in square EK. But if you remember, the 1 in box DJ-FL has to be in either EJ or EK, so the 1 has to be in EJ. So there is a 1 in GL, and now there is a 1 in EJ. There is also a 1 in AO and one in CP. So there has to be a 1 in BK. Now the 1s are done.
 
  J K L M N O P Q R  
  A   9 7   5 1   6    
  B   1 5     6   7    
  C 4     7     1 3 5  
  D 9       1 7   8 3  
  E 1 5   8   3 7      
  F 7 8 3   4     1 2  
  G   4 1     9     7  
  H   7   1     3      
  I   3     7     5 1  
 
Another technique that you can use when the boxes, lines, and columns start getting close to being completed is to figure out what hasn't been filled into that area is. For example, look at line F. There is a 1,2,3,4,7, and an 8. That means that the 5, 6, and 9 are missing. Now look at column O. There is a 6 and a 9. That means that square FO has to be the 5. Now we can put a 5 in square GM, another in square DP, and the final 5 in square HJ. If you didn't follow that, look closely at the columns and rows for all the 5s.
 
  J K L M N O P Q R  
  A   9 7   5 1   6    
  B   1 5     6   7    
  C 4     7     1 3 5  
  D 9       1 7 5 8 3  
  E 1 5   8   3 7      
  F 7 8 3   4 5   1 2  
  G   4 1 5   9     7  
  H 5 7   1     3      
  I   3     7     5 1  
 
Now we can put a 3 in square GN.
 
  J K L M N O P Q R  
  A   9 7   5 1   6    
  B   1 5     6   7    
  C 4     7     1 3 5  
  D 9       1 7 5 8 3  
  E 1 5   8   3 7      
  F 7 8 3   4 5   1 2  
  G   4 1 5 3 9     7  
  H 5 7   1     3      
  I   3     7     5 1  
 
Now look at row F. The 4 at FN forces the 4 in box DP-FR to be in row E in either EQ or ER. That means that the 4 in box DJ-FL has to be in row D. But there is a 4 in column K, so the 4 has to be in square DL.
 
  J K L M N O P Q R  
  A   9 7   5 1   6    
  B   1 5     6   7    
  C 4     7     1 3 5  
  D 9   4   1 7 5 8 3  
  E 1 5   8   3 7      
  F 7 8 3   4 5   1 2  
  G   4 1 5 3 9     7  
  H 5 7   1     3      
  I   3     7     5 1  
 
You might have noticed that the numbers missing from column K are the 2 and the 6, and the numbers missing from box DJ-FL are the 2 and the 6, and the numbers missing from row D are the 2 and the 6. This happens a lot. We can't doing anything with that info now, but at some point there will be a break, and we will be able to enter a whole bunch of numbers.
 
Now look at row G. The numbers missing are the 2, 6, and 8, but there is a 6 and an 8 in column Q. That means that square GQ has to be a 2.
 
  J K L M N O P Q R  
  A   9 7   5 1   6    
  B   1 5     6   7    
  C 4     7     1 3 5  
  D 9   4   1 7 5 8 3  
  E 1 5   8   3 7      
  F 7 8 3   4 5   1 2  
  G   4 1 5 3 9   2 7  
  H 5 7   1     3      
  I   3     7     5 1  
 
Now look at the 9 in row A. It forces a 9 to be in row B in box AP-CR, so there has to be a 9 in either CN or CO. But there is a 9 in GO, so there is a 9 in CN. That means that there is a 9 in FM. That leaves one box in row F, and that's a 6. Placing the 6 in column P means that there is a 6 in square HR, and that means that there is a 6 in square IM and square GJ. That leaves one square in row G, and that is the 8.
 
  J K L M N O P Q R  
  A   9 7   5 1   6    
  B   1 5     6   7    
  C 4     7 9   1 3 5  
  D 9   4   1 7 5 8 3  
  E 1 5   8   3 7      
  F 7 8 3 9 4 5 6 1 2  
  G 6 4 1 5 3 9 8 2 7  
  H 5 7   1     3   6  
  I   3   6 7     5 1  
 
Now look at row D. There is a 2 and a 6 left. But there is a 6 in column M, so the 6 in row D is in square DK. That means that there is a 2 in CK, EL, and DM. And it means there is a 6 in EN and CL. That means that CO is an 8, and then HN is an 8. There is an 8 in AJ or AB, and there is an 8 in HN, so another 8 is in IL.
 
  J K L M N O P Q R  
  A   9 7   5 1   6    
  B   1 5     6   7    
  C 4 2 6 7 9 8 1 3 5  
  D 9 6 4 2 1 7 5 8 3  
  E 1 5 2 8 6 3 7      
  F 7 8 3 9 4 5 6 1 2  
  G 6 4 1 5 3 9 8 2 7  
  H 5 7   1 8   3   6  
  I   3 8 6 7     5 1  
 
That means that there is a 9 in HL and a 2 in IJ. So there is a 2 in HO. That means there is a 4 in IO.
 
  J K L M N O P Q R  
  A   9 7   5 1   6    
  B   1 5     6   7    
  C 4 2 6 7 9 8 1 3 5  
  D 9 6 4 2 1 7 5 8 3  
  E 1 5 2 8 6 3 7      
  F 7 8 3 9 4 5 6 1 2  
  G 6 4 1 5 3 9 8 2 7  
  H 5 7 9 1 8 2 3   6  
  I 2 3 8 6 7 4   5 1  
 
Now you can see that there is a 9 in IP and a 4 in HQ, and therefore there is a 9 in EQ and a 4 in ER.
 
  J K L M N O P Q R  
  A   9 7   5 1   6    
  B   1 5     6   7    
  C 4 2 6 7 9 8 1 3 5  
  D 9 6 4 2 1 7 5 8 3  
  E 1 5 2 8 6 3 7 9 4  
  F 7 8 3 9 4 5 6 1 2  
  G 6 4 1 5 3 9 8 2 7  
  H 5 7 9 1 8 2 3 4 6  
  I 2 3 8 6 7 4 9 5 1  
 
There is a 2 in BN and in AP. That means that there is a 4 in BP and in AM.
 
  J K L M N O P Q R  
  A   9 7 4 5 1 2 6    
  B   1 5   2 6 4 7    
  C 4 2 6 7 9 8 1 3 5  
  D 9 6 4 2 1 7 5 8 3  
  E 1 5 2 8 6 3 7 9 4  
  F 7 8 3 9 4 5 6 1 2  
  G 6 4 1 5 3 9 8 2 7  
  H 5 7 9 1 8 2 3 4 6  
  I 2 3 8 6 7 4 9 5 1  
 
So you can see that BM is a 3 and so is AJ. There is an 8 in BJ and AR. And finally there is a 9 in BR.
 
  J K L M N O P Q R  
  A 3 9 7 4 5 1 2 6 8  
  B 8 1 5 3 2 6 4 7 9  
  C 4 2 6 7 9 8 1 3 5  
  D 9 6 4 2 1 7 5 8 3  
  E 1 5 2 8 6 3 7 9 4  
  F 7 8 3 9 4 5 6 1 2  
  G 6 4 1 5 3 9 8 2 7  
  H 5 7 9 1 8 2 3 4 6  
  I 2 3 8 6 7 4 9 5 1  
 
We are done. As you can see, the numbers come fast and furious at the end.
 
 
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