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Making Use of Counting

Let's have a look at a puzzle which has had all of the possible numbers counted out and written down:

 
The very simplest thing we can do is look for squares where there is only one possible number which can go there. When you find a square like that, fill it in with the one possible number you've found. In the puzzle below, the green square at [6,2] has only one solution.
 
Don't stop there though! You've shown that [6,2] must hold an 8, and that means that none of the other squares in that row can hold an 8, nor can any squares in the same column, nor can any squares in the same 3x3 box. In the puzzle below, we've marked all the squares which can have their 8s crossed out in pink:
 
There's often a flow-on effect. Once you've crossed out the 8 in square [4,2], the only possibility is 7, so fill that in. Then cross out all other 7s in the "possibles" sections of the row, column and 3x3 box for square [4,2]. This will mean you cross out the 7 in square [7,2], which then only holds a 1. And so on...

The technique we've just been using takes advantage of the fact that a number can only occur once in a given row, column or 3x3 block. The next technique takes advantage of the fact that each number from 1..9 must occur once in each row, column, or 3x3 block. This means that if a number occurs only once in a particular row (or column, or 3x3 box), then that must be the correct number, even if there are other "possibles" written in the same square. Have a look at square [2, 3] in the puzzle below:

If we look at column 2 (highlighted in pale green) we can see that the number 4 occurs in only one square, [2, 3] (highlighted in dark green). This means that this square must contain 4, despite having more than one possibility written down. Fill the square in with 4, and don't forget to cross out any other 4s you find in the same row, column, or 3x3 box. In this case, square [1, 3], immediately to the left, has a 4 which can now be eliminated, as we know that any number can occur only once per row.

 

Next we'll be looking at quite a high powered technique, using pairs and triplets of numbers to eliminate possibilities elsewhere on the board, in Tutorial 4.