Francis’ Rules for Sudoku

The book “Sudoku - Explaining the fifteen steps of Francis’ Rules for Sudoku” was first published in 2006. Since then, another book “Sudoku is Fun: Taking the Francis’ Run” was published later in the same year. The preface to the first book reads thus:

“If you are into solving Sudoku challenges, this book should be read. It illustrates how you can apply Francis’ Rules to shorten the time taken to obtain a solution to any Sudoku challenge. 

“If you are not into Sudoku Challenges, then you are missing out on an opportunity for the development of your brain power to meet the needs of modern-day society. More importantly, Sudoku can assist you in overcoming memory loss. Sudoku can easily be done during the many waiting periods you have while boarding your flight, waiting to be served in a restaurant or while on a train. 

“This book tells you how you can start exercising your mind with the logical steps of a modern Sherlock Holmes. “

The fifteen rules of the Francis’ Rules are divided into five groups of three rules each.

Before we re-state the fifteen rules, it is important to say that the Rules are designed for those who are either new to the Sudoku Challenge or for those who have in the past being solving the Sudoku Challenge in a haphazard manner, hitting numbers into vacant cells without any logical methodology. The fifteen steps provide us with a logical pattern to solve a Challenge.

It is important for beginners to realize that a Sudoku Challenge is one where we insert numbers 1 through to 9 in each vacant cell such that the numbers in each row, in each column and in each box such that the numbers are unique for that row, that column and that box. At the start, there are some cells have numbers already given to us as clues to fill up the other vacant cells.

For ease of reference we name the columns column “a” right through to column “i” and the rows as row “1” right through to row “9”.

The nine boxes each comprises nine cells and numbering them from left to right, top to bottom, the boxes are:

Box # 1: cells a1, a2, a3, b1, b2, b3 and c1, c2 and c3;

Box # 2: cells d1, d2, d3, e1, e2, e3 and f1, f2 and f3;

Box # 3: cells g1, g2, g3, h1, h2, h3 and i1 i2 and i3;

Box #4: cells a4, a5, a6, b4, b5, b6 and c4, c5 and c6;

Box # 5: (What are the cells here?)

Box # 6: (What are the cells here?)

Box # 7: cells a7, a8, a9, b7, b8, b9 and c7, c8 and c9;

Box # 8 and Box # 9: (What are the cells here in these two boxes?)

Now let us look at the systematic and logical approach to solving a Sudoku Challenge under the Francis’ Rules.

Group A – Cross Referencing

Rule # 1: Look for the correct number along the three adjacent rows of the three adjoining horizontal boxes.

For example, looks at the horizontal rows in sets of three, corresponding to the boxes. We call this Rule the Cross Referencing of the Rows. In discussing the Francis’ Rules, a Sudoku challenge is looked at from the viewpoint of a spreadsheet with 9 cells horizontally by 9 cells vertically. The columns are named from A to I and the rows are named 1 to 9 respectively. The 3 cells x 3 cells boxes are named Box 1 (cells A1 to C3); Box 2 (cells D1 to F3); Box 3 (cells G1 to I3); Box 4 (cell A4 to C6) and so on until Box 9 is represented by cells G7 to I9.

Let us take the Sudoku Challenge below as an example to illustrate this Rule # 1. Looking at rows 1, 2 and 3 in Box # 1, 2, and 3, we have cell c1 = cell d2 = 2. Therefore using Rule # 1, cell i3 = 2. Similarly, cell h1 = cell c2 = cell d6 = 1. Therefore using Rule # 1 again, cell f3 = 1. Then cell a1 = cell g2 = 4, so cell d3 = 4.

Now we look at rows 4, 5 and 6 and using Rule # 1 again, we have cell a5 = cell i6 = 3. So cell f4 = 3.

On to rows 7, 8 and 9 and using Rule # 1 again, we have cell e7 = cell i8 = 4, so cell b9 = 4. Then cell c8 = cell g7 = f2 = 9, so cell d9 = 9.

Now perhaps it is time to give the rest of the fifteen Francis’ Rules and let you the readers to figure out the solutions as this article is going perhaps too long. We shall consider how to apply all the fifteen Francis’ Rules in future postings.

Rule # 2: Look for the correct number along the three adjacent columns of the three adjoining vertical boxes.

Look at the three columns collectively, Columns A, B and C, then Columns D,E and F and finally G, H and I for the correct number in a cell, reflecting upon the other numbers along rows and boxes that can affect the result.

Rule # 3: Look from the viewpoint of a particular cell into numbers of the same row or of the same column or of the same box and see that only one number fits that particular cell.

Group B – Remaining Digit

Rule # 4: Look at every box where there are three or less remaining cells to be filled in. In that case, it may be possible for us to deduce which remaining number that can fit into which remaining cell.

Rule # 5: Do the same as Rule # 4 to every row.
Rule # 6: And the same to every column.
The next three rules are collectively known as the Pairing Rules.

Group C – Pairing

Rule # 7: For any box, there may be instances where two similar numbers can sit comfortably into two cells in that box. By a process of eliminating these numbers from the rest of the remaining cells of that box, the other cells can be filled with the correct numbers.

Rule # 8: Similarly, we can do the same pairing for any row.

Rule # 9: And for any column.

Group D – Only Digit Rules.

Rule # 10: For any box, we can sometimes find that external conditions are such that only one number fits into just one cell in that box.

Rule # 11: Similarly, we can find that only one number that can fit into just one cell in a row.

Rule # 12: And similarly, we can find that only one number fits into just one cell in a column,

Group E – Triple Digits Rules.

Rule # 13: Just like the pairing rules (Group C) we can find that within a box there can be three similar numbers sharing three cells. When that happens, eliminating these three numbers from the those cells can result in definite numbers for the remaining cells of the box.

Rule # 14: In a similar manner we may have three numbers sharing three cells in a row, thus giving answers to the remaining cells of that row.

Rule # 15: Again, this can happen to three cells in any column.