Is this Sudoku really HARD? 这个数独真的很难吗？

Let us revise the fifteen Francis’ Rules for Sudoku.

The first three rules are called “Cross Referencing Rules”. We take a good look at each Row, each Column, and each Cell to see whether there are two similar digits in two cells of a Row (Rule # 1) or of a Column (Rule # 2). Then examine each cell in the light of the digits in the surrounding Rows, Columns, or Squares to see if there is just one number that can fit into a particular cell (Rule # 3).

The second three rules are collectively called “Remaining Cell”. When in a Box, there are less than four uncovered digits, those digits can be easily found in a mental process of elimination of numbers (Rule # 4). In a similar manner, the remaining Cell can be applied to a Row (Rule # 5) or to a Column (Rule # 6).

The next three rules concern the pairing of two cells in a Square each with the possibility of two similar digits (Rule # 7), or in a Row (Rule # 8) or in a Column (Rule # 9). If a pair is found, then the digits for the other cells in that square, or that row, or that column can easily be found.

Rules # 10, # 11, and # 12 deal with situations when only one digit is possible in a particular cell found on a box (Rule # 11), or in a Row (Rule # 12) or in a Column (Rule # 12).

The last three rules are collectively called “Tripling three cells”. These rules apply in the same manner as in Rules # 7, # 8, and # 9, except that this time, there are three possible numbers in three cells of a Square (Rule # 13), or of a Row (Rule # 14) or of a Column (Rule # 15).

NOTE: The Chinese version of this article will be posted in another later webpage. 请注意：本文的中文版将在稍后的另一个网页中发布。

The Hard Sudoku today from the NZ Herald is shown above.

We start solving this Sudoku Challenge by applying Francis’ Cross Referencing Rules. Looking at Rows # 1, # 2, and # 3, we find cell c2 = i2 = 2 = f9. Therefore cell e1 is equal to 2. [Francis’ Rule # 1]. Step # 1.

Likewise, in Rows # 7, # 8, and #9, we have cell a7 = cell g9 = 9 and is also the digit for cell d6. So cell f8 = 9. [Francis’ Rule # 1]. Step # 2.

We look now at Francis’ Rule # 2 now, looking at the columns, three at a time. In Columns g, h, and I, we find that cell g4 = cell h3 = 6 and is also equal to the same digit for cell c8. Therefore cell i9 is equal to 6 as well. [Francis’ Rule # 2]. Step # 3.

At Row # 5 now, we find three cells without digits and the missing digits are 4, 7, and 9. We then notice that in Column a, cell a3 = 4 and cell a7 = 9. Therefore the digit for cell a5 must be the remaining digit 7. [Francis’ Rule # 5]. Step # 4. Sticking on to Row # 5, we can easily deduce that cell e5 = 4 (Step # 5) because cell e2 = 9; and the remaining cell i5 = 9. [Francis’ Rule # 5]. Step # 6.

Then for Columns a, b, and c, we have cell a5 = cell c1 = 7 = cell g8. So, cell b9 = 7. [Francis’ Rule # 2]. Step # 7. Likewise, for Columns g, h, and I, we can logically deduce that cell h1 = 9 (Step # 8) because cell g9 = cell i5 = 9 = cell e2. [Francis’ Rule # 2].

At Box # 1, we look at the cells surrounding cell b1, and if we run our finger to every exposed digits in the relevant columns and rows, we find that cell b1 = 8. [Francis’ Rule # 11]. Step # 9.

If we now pair 5 and 9 for cells b3 and c3 in Row # 3, we have cell g3 = 3. [Francis’ Rule # 8]. Step # 10.

At Box # 3, we look at the cells surrounding cell g1, we find that only digit 4 fits into this cell. So cell g1 = 4. [Francis’ Rule # 10]. Step # 11. Repeat this again for cell i1 on Box # 3, we get cell i1 = 1. [Francis’ Rule # 10]. Step # 12.

In Row # 1, we have two remaining cells for digits 3 and 6. But, cell f4 = 3, therefore cell f1 must equal to 6. [Francis’ Rule # 5]. Step # 13. The last remaining cell in Row # 1, cell a1 must equal to 3. [Francis’ Rule 5]. Step # 14.

In Box # 1, we can pair 5,9 in cells b3 and c3, and with cell a1 =3, we know that cell a2 = 6. [Francis’ Rule # 7]. Step # 15.

We turn our attention now to rows # 1, 2, and 3 and notice that cell a1 = cell g3 = 3 = cell f4. Using Francis’ Rule # 1, therefore, cell d2 = 3. Step # 16. Similarly and also looking at rows # 1, 2, and 3, we have cell a3 = cell g1 = 4. Therefore cell f2 = 4. [Francis’ Rule # 1]. Steps # 17.

We now look at Columns g, h, and I. Here, cell g3 = cell h8 = 3 = cell f4. So, cell i6 = 3. [Francis Rule # 2]. Step # 18. For Column f, there are three remaining cells now for digits 1, 7, and 8. Since cell i7 = cell c6 = 8, cell f3 = 8. [Francis’ Rule # 6]. Step # 19.

For Rows # 7, # 8, # 9, we have cell b7 = cell h8 = 3 = cell d2. Therefore cell e9 = 3. [Francis’ Rule # 1]. Step # 20.

Similarly, for Rows # 4, # 5, and # 6, we find that cell b5 = cell g4 = 6 = cell f1. So logically cell e6 = 6. [Francis’ Rule # 1] Step # 21.

At Columns d, e, and f, we have cell f1 = cell e6 = 6 =cell c8 = cell i9. The digit for cell d7 is 6. [Francis’ Rule # 2]. Step # 22.

For Column e, there are three digits remaining 1, 7, and 8 for three vacant cells. Cell f3 = 8 and is equal with cell i3. Therefore cell e4 = 8. [Francis’ Rule 6]. Step # 23.

Step # 24, looking at Column i there is only one digit (4) that can fit into cell i8. So cell i8 = 4. [Francis’ Rule 12].

For Row # 8, there are three cells remaining for three digits of 1, 2, and 8. As cell d5 = 2; and cell d3 or d4 = 1, cell d8 = 8. [Francis’ Rule # 5]. Step # 25. The remaining two digits for Row # 8 are now 1, and 2. With cell b2 = 1, cell b8 = 2. (Step # 26). The remaining cell a8 in Row # 8 is equal to 1. (Step # 27).

For Column d, if we pair 1, 7 in d3 and d4, we can discover that cell d9 = 4. [Francis’ Rule # 9]. Step # 28.

Looking at Rows # 7, 8, and 9, we have cell d9 = cell i8 = 4, and therefore cell c7 = 4. [Francis’ Rule # 1]. Step # 29.

In Row # 6, if we pair 2, 5 in cell a6 and cell g6, we have cell b6 = 4. [Francis’ Rule # 8]. Step # 30.

Looking at Rows 4, 5, and 6, we can see cell b6 = cell = e5 = 4 = cell i8. So, cell h4 = 4. [Francis’ Rule # 1]. Step # 31. Then looking at Columns g, h, and I, we have cell i1 = cell g5 = 1 = either cell e7 or cell f7 and therefore cell h9 = 1. [Francis’ Rule # 2]. Step # 32.

The two remaining cells of Row # 9 can now be deduced. Cell c9 = 5 and cell a9 = 8. [Francis’ Rule # 5]. Steps # 33 and 34.

Francis’ Rule # 2 can be applied to find cell e4 = 1 : cell b2 = cell a8 = 1. (Step # 35).

The next five steps # 36 to 40 concern the last remaining cell for their respective categories - Column c, cell c3 = 9 [Francis’ Rule # 6]; Box # 1, cell b3 = 5 [Francis’ Rule # 4]; Column b, cell b4 = 9 [Francis’ Rule # 6]; Column d, cell d4 = 7 and cell d3 = 1 [Francis’ Rule # 6].

The logical steps from # 41 through to # 53 are tabulated below. The solution to all the fifty-three cells can be seen in the photograph of the final Sudoku grid. Most of the cells are those of the remaining cells for Boxes # 6 and # 4; for Rows # 3, # 4, # 6 and # 7; for Columns e, g, h, and i