#Method of solving sudoku puzzles how to#

We select the set of pairs (the lists of length two).ģ. This obvious fact is surprisingly useful in solving Sudoku puzzles.ġ. Hence, no other entry in that row, column or block can contain either or. If any row, column or block contains the pair twice, both and must be used up in the two entries containing, even though we do not know which pair contains and which contains. We are done! We explore this technique further in the next section. Then we use again and display the result. Similarly, in the first row, there are three entries that are sublists of, so we remove 5, 6 and 8 from at the end of row 1 this defines. While we do not know the exact value of any of the red entries, we know that the three numbers 5, 6 and 8 will be used up filling them thus we can remove 5, 6 and 8 from the other entries in this block (colored green). However, the first block has three entries (colored red) that are all sublists of. The puzzle simplifies, but we see that we are still not done! To apply again and again to until the result no longer changes, we use. įinally, if is a list of one element, we redefine it to be that element. If the entry is a list rather than an integer, we redefine by removing the entries that also belong to. Since no entry can appear more than once in any row, column or block, we let be the set of integers in the row, column and block containing. We start with in order to be able to redefine entries. Our next task is to start eliminating candidate values in the entries that are lists of numbers in X, proceeding one entry at a time. Our first step is to replace each □ in (using ) by the list of the nine numbers,, which are possible candidates to occupy that position in. To get a single list of these entries by removing the inner parentheses, we use. To do this we define a function that gives a list of the entries that comprise the block of in X.įor example, this is the block containing, the sixth entry in row 1, □. It is more difficult to obtain the block to which belongs. The function displays that list vertically. For example, the column that contains, that is, column 5 of, can be obtained by entering. However, Mathematica has a faster way using the option. One could first “transpose ” the matrix, that is, interchange rows and columns and then find row of the transposed matrix the command for transposing a matrix is. To obtain the column in X that contains is a little trickier. To obtain the entire row in X that contains the entry, we evaluate so, for example, this gives row 3, which contains element. In attempting a solution, a blank gets replaced with a list of candidate entries shown compactly without braces or commas.Įach element of a Sudoku matrix is obtainable as, where and are the row and column of the element in the matrix.

We can also display this in Sudoku format by drawing column and row lines and a frame. A list of lists of the same length is regarded as a matrix in Mathematica, so we input for the puzzle and then show it in matrix form. The input for this puzzle is a list of nine lists consisting of blanks (shown as □) or integers between 1 and 9. Some of the entries are already chosen, and the final puzzle solution must contain these initial choices. The 81 entries are to be filled with the integers 1 to 9 in such a way that each row, column and subgrid contains all the nine integers.

Sudoku, for those unfamiliar with this puzzle, consists of a square grid with nine subgrids. The programming skills required are kept at a minimum. The hardest puzzles require some guessing, and we include a simple backtracking technique that solves even the hardest puzzles.

#Method of solving sudoku puzzles how to#

We simultaneously introduce effective techniques for solving Sudoku puzzles and explain how to implement them in Mathematica.