# How to Solve Sudoku Puzzles

One of the latest popular leisure time activities is dong Sudoku puzzles. It’s a lot of fun. Every puzzle is solvable by everyone. It just takes patience by breaking the puzzle down into parts.

Sudoku puzzles can take several forms. The most common puzzle is a 9×9 block of squares. Some puzzles are 16×16. Let’s look at the 9×9 to start.

The puzzle starts with nine rows of nine squares. Some of the squares have numbers in them already. There are also dark lines that mark where nine 3×3 blocks of squares are. The object of the game is to fill each square with a number so that each row has the numbers 1 through 9, each row has the numbers 1 through 9, and each 3×3 block have the numbers 1 through 9.

Remember that each number occurs only once in each column. So, if a number appears once in a column, then that number can be eliminated as a possibility for all of the other squares in that column. Similarly, each number appears only once in each row, and in each 3×3 block of squares. Make an enlarged copy of the puzzle so that notes can be put into each square. Now, let’s begin.

ELIMINATING TWO ROWS AND TWO 3×3 BLOCKS AT A TIME

The top three rows also form the top three 3×3 blocks of squares. Look to see if there are two occurrences of the number “1” in the top three rows. If there is, notice that the “1”s are in separate rows, and that they are in separate 3×3 blocks of squares. Mentally, cross out the two rows that have “1”s and cross out the two 3×3 blocks that have the “1”s. There must be a “1” in the third remaining 3×3 block, and it has to be in the remaining third row. That means that there are only three squares where the “1” can be. If two of those three squares are already filled with numbers, then the “1” must occur in the remaining square, so mark it in. If more than one of the three remaining squares are blank, then look down the columns for each of the three squares. If a “1” appears in one of those columns, then the “1” cannot occur in the blank square that we were checking because only one “1” can be in each column. If two of the three squares can be eliminated in this way, then the remaining square must have the “1”. If the location for the “1” can not be isolated, go on to trying to do the same for the middle three rows, which form the middle three 3×3 blocks of squares. In turn, check the bottom three rows, which contain the bottom three 3×3 blocks of squares. When that is done, start looking down the first three columns, then the second block of three, then the third block of three. When the search is done for “1”s, then do the same for each of the other numbers.

FILL IN ALL OF THE POSSIBILITIES FOR EACH SQUARE

Now that the “easy” squares have been filled, some tedious, but fruitful work needs to be done.
Find a blank square and fill in all of the numbers that have a chance of occurring in that square. To do that, see if the number “1” occurs elsewhere on that row, on that column and in that 3×3 block of squares. If a “1” does not occur in any of those squares, then mark a “1” in the blank square. Now do the same for the other numbers “2” through “9”. Each square will have several possible numbers listed, sometimes five or six of them. Now do that for every blank square in the puzzle. This may take a while, but it is good practice in paying attention. Putting an incorrect number in a square as a possibility would be bad. In some cases, there will only be one number that is possible for that square. In that case, mark that number in that square boldly.

LOOK FOR SINGLE POSSIBILITIES

Look down the first row to see how many times a “1” is possible. It has to be possible at least once. If a “1” is possible in only one square of the row, then mark that square as having a “1” even if other numbers were listed as possibilities in that square. Cross the other numbers out! Do the same for each row, then for each column, then for each 3×3 block of squares. When the search for “1”s are complete, then search for the other numbers.

ELIMINATING NUMBERS IN A 3×3 BLOCK

Suppose that a number, say “4”, is possible in several locations in a 3×3 block, but on one of the rows that has “4”s, there are no “4”s outside of that 3×3 block. So, that means that a “4” must occur in that row of the block. It also means that none of the other “4”s in that block are possible, so they can be crossed out. Of course, the same can be true in searching for other numbers, and it is true for rows as well as columns.

PAIRS OF PAIRS

Suppose that a square has a “2” and a “4” as the only possibilities, and another square in that row also has a “2” and “4” as the only possibilities. That means that if one square is a “2”, then the other square must be a “4”. So, any other “2”s and “4”s on that row can be crossed out. The same is true for pairs of pairs in the same column, or 3×3 block.

RE-RUN

Every time that square is filled with a number, go through that row, column and block, and cross out that number. Once every row, and every column, and every 3×3 block have been checked, start checking all over again. Whenever a square is filled, the possibilities change, so it is worth going through what is left of the puzzle again.

ORGANIZED GUESSING

Many puzzles can be solved using the previous techniques. If the puzzle still isn’t solved, one can resort to limited guessing. Suppose a square has only two possibilities, say a “5” and a “9”. Make a guess as to what the value is, say a “5”, but write down what the guess was on the side and the square where the first guess was made. Mark the guess with a circle around the number. Using this way makes it possible to distinguish between the guess and the certain values. Continue to deduce values for the squares, but put circles around the values that are derived from the guess. If the guess is wrong, then eventually, a contradiction will occur. Perhaps it will look like a number occurs twice in a row. That means that the guess was wrong. So, erase all of the numbers with circles. However, this also means that the other number, the “9” in this case was really correct. So, mark that number, the “9”, boldly.

SECOND GUESSING

In some really hard puzzles, one might have to make a second guess. So, make a triangle around the second guess and record where the guess was made. Now continue deducing values. Mark each number that was derived from the second guess with a triangle. In this way, if a contradiction is reached, one can erase the number with triangles, but leave the numbers with circles there. If both choices for the second guess reach contradictions, then that means that the original guess, the circles, are wrong. So, in that case, erase all triangles and circles. But that’s okay, because if the original guess was wrong, then the other possibility was correct. If a puzzle is completed with a guess, then the guess was correct after all.

OTHER PUZZLE VARIANTS

The same principles apply to other Sudoku variants such as the puzzles where the diagonals must have “1” through “9”. That restriction provides more reasons to eliminate possibilities. If a number occurs once in a diagonal, then it cannot occur elsewhere in the diagonal, or in the row, or in the column, or in the 3×3 block. For the 16×16 puzzles, use single hexadecimal representations: “0” through “9” and “A” through “F” as the sixteen numbers.
So, have fun with Sudoku and good luck!