Solving Sudoku: Advanced Strategies

As sudoku puzzles advance to more difficult levels, more advanced strategies are required to solve them.  While basic strategies will get you started with a more advanced puzzle, they will not be enough to get you to the finish line.  Fortunately, advanced strategies often build off basic strategies.

• Single’s Chain – A single’s chain builds from the basic strategy of conjugated pairs.  Whether the pairs are hidden or naked makes no difference.  The basic idea is to form a chain between pairs that could contain the same digit.  Example: The number you are focusing on is 6.  In the fourth row, there are two squares that could be 6.  These squares are A and B, and they form a pair.  B also forms a pair with C, which is in the same column.  C forms a pair with D, which is in the same block. D then forms a pair with E, which is in the same row.  E is also in the same column as A, so E and A are pairs. If you were to draw a line from square to square, you would be able to see the chain.  How does this help?  Thinking of the rules of sudoku, if A is 6, then B cannot be 6. If B is not 6, then C is 6, which means that D is not.  By the same token, if B is 6, the A and C cannot be.  If you look at the chain again, you will notice that D and E are pairs, and A and E are pairs.  If A is 6, then following the chain will lead you to D is not 6, which means E is.  E and A cannot both be 6 because they are in the same column. This brings you to the first rule of single’s chains.
• If one square relates to two alternate squares by row, column or block, then the square is excluded.  In this case, E relates by column to A and by row to D.  If you look at the chain, D is not 6 if A is 6, so they are alternate squares.  E can be excluded because it relates to both D and A, which are alternating squares.
• This can be made easier by using colors. A and C would be red, while B and D would be blue.  When you look at E, it’s in the same column as a red square and the same row as a blue square.
• The second rule: If there are two candidates in the same unit, or two of the same colored square when working with colors, then they are false squares.  In this case, coloring in pair order would mean that you colored A red, then colored B and E blue because they are A’s pairs.  C would then be colored red, and D would be colored blue.  D and E are in the same row, and they are both blue.  This means that blue is the false color, so those in red, A and C, get the number 6.
• Y-Wing – This is actually much simpler than it often sounds, and builds somewhat from pairs.  Assume that your sudoku puzzle has a empty squares that form a rectangle shape.  In each of the four corners, there are number candidates.  For example, A and B are in the same row, while C and D are in the same row.  A and C are in the same column, as are B and D.  Corner A contains either 4 or 7.  Corner B contains either 7 or 9.  Corner C contains either 9 or 4.  If corner A contains 4, then corner C contains 9.  If corner A contains 7, then corner B contains 9.  In either case, either B or C will contain 9.  Corner D shares a row with corner C and a column with corner B.  Since D is in the same unit with both, 9 cannot be in corner D.  How does this help?  While the Y-Wing may not tell you which number is in which square, it does tell you what is not in corner D.  That elimination could open the door to one of the other sudoku strategies, and solving an entire row, column or block.