Try Solving This Sudoku Challenge
This morning let us try to solve this Sudoku Challenge.
There are thirty-five exposed cells here out of a total of eighty-one cells.
Your challenge is to fill the rest of the forty-six blank cells with numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9 such that:
(a) Each row (named row 1 to 9) contains the nine digits 1 to 9 with no repeated digit;
(b) Each column (named column a to i) contains the nine digits 1 to 9 with no repeated digit, and
(c) Each 3 x 3 box (named box 1 to box 9) contains the nine digits 1 to 9 with no repeated digit. [The boxes are colour-shaded for ease of identifying and the boxes are named from left to right, top to bottom).
A sudoku challenge is a grid of nine by nine cells, totaling eighty-one cells. The grid has been subdivided into nine subgrids or “boxes” of three by three cells for each box.
The challenge begins with some cells already exposed with digits 1 to 9 in them. In general, the more exposed cells there are, the easier the challenge. Any challenge that has less than twenty-six exposed cells can be generally classified as ‘impossible’ to solve and thus you will be wasting your time trying to solve such a challenge. For beginners, try to solve challenges that contain more than thirty exposed cells.
To repeat the objective of Sudoku again, each cell has to contain one digit from 1 to 9 in such a way that:
each cell in a horizontal row has one digit each once; and
each cell in a vertical column has one digit each once; and
each cell in a three-by-three box has one digit each once.
To solve a Sudoku Challenge, I have developed fifteen rules called “Francis’ Rules for Sudoku”. The first six rules are:
Rule 1 – Cross-Referencing in three adjacent rows.
This rule requires us to look horizontally across three rows with three boxes – Boxes 1, 2 and 3 OR Boxes 4, 5 and 6 OR Boxes 7, 8 and 9 – to see that there are no two similar digits within a row, a column or a box.
Rule 2 – Cross-Referencing in three adjacent columns.
This rule is similar to Rule 1 except we have to look at the boxes in a vertical manner – Boxes 1, 4 and 7 OR Boxes 2, 5 and 8 OR Boxes 3, 6 and 9 – to see that there are no two similar digits within a row, a column or a box.
Rule 3 – Cross-Referencing Rows, Columns, and Boxes.
This Rule asks us to look at the row, the column and the box for any single cell to see if there can only be one permissible digit that sits in that cell.
Rule 4 – Remaining Digit(s) in Box.
This rule is applicable where there are three or less vacant cells remaining to be filled within a Box. We can easily deduce the remaining digits by a process of elimination.
Rule 5 – Remaining Digit(s) in Row.
This rule is similar to the last except that instead of a box, we are looking for remaining three or less vacant cells remaining to be filled within any Row.
Rule 6 – Remaining Digit(s) in Columns.
Likewise, if there are three or less remaining vacant cells remaining to be filled within a Column, this rule can give meaning answers to these vacant cells.
Armed with these inital six simple logical rules, you can easily start to fill in the following cells:
Cell c4 = 8; Cell c9 = 1; Cell d8 = 6; Cell a8 = 2; Cell a6 = 6; Cell c1 = 4; Cell b3 = 5; Cell f7 = 8.
Then looking at box 1 (cells a1, a2, a3, b1, b2, b3, c1, c2, c3) we can see that the remaining digits 1 and 3 can be allocated to cell c1 = 1 and cell b2 = 3 as there is a digit 1 in column b in cell b4.
In a similar manner, if we look at box 4 (cells a4, a5, a6, b4, b5, b6, c4, c5 c6) we can deduce that cell c5 = 3 and cell b6 = 7.
Then if we look at columns a and b, we can see that cell a7 = 9 and cell b9 = 8.
How about row 7 where the remaining digits are 3 and 7? Can you see that cell d7 = 7 and cell h7 = 3? Can you now see that cell i9 = 7? And, subsequently, cell g5 = 7?
Is it easy so far?
Now for Francis’ Rules #6 to 9:
Francis’ Rule # 7, Francis’ Rule # 8 and Francis’ Rule # 9 cover the necessity for pairing two cells within a Box (Rule # 7); or a Row (Rule # 8) or a Column (Rule # 9).
Another unsaid rule is this. Because we have all the digits 1 through to 9 in every row, every column, and every box, the digits should add up to a sum of 45 for each row, each column, and each box.
Pairing rules like Rule # 7, Rule # 8 and Rule # 9 involve a decision to have two cells in one Box, Row or Column to have the same pair of digits. Once a decision is made, the other remaining cells in that Box, Row or Column can easily be determined. However, we need to be prepared for the tracing back to this decision should the filled figures fail to match up to a total sum of 45 in any box, any row or any column.
Try to use Rules # 7 to 9 to solve the remaining exposed cells in this Challenge.
It is really simple!
https://www.amazon.com/Sudoku-Explaining-Fifteen-Steps-Francis/dp/1412080231