Teach Yourself Sudoku

Learn the secrets that let you easily solve Sudoku puzzles faster!

Lesson #6A. Check for BLOCK OUTS


BLOCK OUTS are based on the simple idea that if you have more than one occurrence of a number, it becomes a lot easier to determine where the other occurrences of that number will go.  In its most commonly used implementation, you look for 2 occurrences of a number in ROW+BLOCK or COLUMN+BLOCK and try to place the third occurrence.


What’s a ROW+BLOCK or a COLUMN+BLOCK? Good question.  They are rectangular 3 by 9 areas running either horizontally or vertically.  Each Sudoku puzzle has 3 of each.  The easiest way to understand where they are is to look at the illustrations below.





Simple, no?  Well, the good news is that something this simple can sometimes help you to quickly resolve the numbers in lots of cells. (Note: the technique you are about to learn can be used at any time during the process of solving a Sudoku puzzle.  Sometimes you can use it right away and other times, you’ll need to resolve a few cells before it becomes useful.  In either case, it’s solving technique that you’ll definitely need in your toolbox and trust me, you’ll use it in every Sudoku puzzle you play.  Most often, easy to moderate puzzles will automatically have at least one instance where this technique can be immediately applied so be sure to take advantage of those opportunities so that you’ll get off to a quick start.)


Take a look at this COLUMN+BLOCK fragment and give particular attention to where the number 7 appears.



The 7s are in the second and third Columns and middle and bottom Blocks.  Given that a number may appear only once in any given Row, Column, or Block, that means that the last of the three 7s for this COLUMN+BLOCK, must be placed in the upper Block (contains the numbers 1-4-9.)  And not only that, it must be placed in the first Column.


But when we look in the first Column, we see that 2 of the 3 cells are already occupied by the numbers 1 and 9.  Therefore the only place left to put the third 7 is in the cell between the 1 and 9 in Column 1.


So without having to count from 1 to 9 numerous times and writing down all of the relevant POSSIBLES, you’ll be able to quickly scan for two occurrences of any number in a ROW+BLOCK or COLUMN+BLOCK and perhaps place the third occurrence immediately.  I saw “perhaps” because you don’t always have the situation where two of the possible cells are already filled as we did in this example.  In that case, you know that the number belongs in a particular ROW / COLUMN within a particular BLOCK but you’ll need to solve a few more cells before you can place the number definitively.


Here’s an example of using BLOCK OUTS in both Rows and Columns.


Take a look at the Sudoku puzzle fragment below paying particular attention to the number 4 in the leftmost COLUMN+BLOCK.



There are two 4s in this COLUMN+BLOCK (in rows 5 and 9 respectively.)  When we block out the corresponding Columns, we are left with three Cells that could hold the third 4.  The first two are blank and the third one is already occupied with a 3.  So which cell gets the 4?  I have no idea and neither should you unless….you looked over to the right into the adjacent COLUMN+BLOCK where lo and behold, the first row contains another 4 which blocks it out.


Our annotated grid fragment now looks like this:



Now we have one and only one Cell that can hole 4, in the second row, immediately above the 3.


So by combining BLOCK OUTS in both COLUMNS and ROWS, you can synergistically leverage the contributions of each to quickly resolve a cell or two with a minimum of effort.



NEXT UP: Lesson #6B.  More BLOCK OUTS


Copyright 2006 Gary Ward All Rights Reserved