Logical Sudoku 逻辑数独

If we were to read this article on the algorithm for solving a Sudoku puzzle written by J. F. Crook, we would fall asleep by the third or fourth page of the nine-page article. Click on to this link to see if that is true for you. It is better that we plunge ourselves straight into solving a puzzle. But do bear in mind that there can be times when no solution is possible, or worse, more than one solution to a puzzle can also be possible. (https://www.ams.org/notices/200904/tx090400460p.pdf)

The aim of a sudoku puzzle is to complete every cell in the puzzle grid with a number (between 1 and 9, no zero) such that all the nine numbers with no repeated number are present in each row, each column and each 3 cells x 3 cells box.

Let us look at this puzzle. There are twenty-six numbers exposed to us. If there are less than twenty-six exposed numbers, our advice is not to attempt doing that puzzle as it is likely that we need to make assumptions later into the solution and that can take a considerable time to obtain a solution.

There is a book entitled “Explaining the fifteen steps of the Francis’ Rules for Sudoku”, published in 2006 that can be helpful. We shall use these fifteen Rules in our logical quest for an answer to the above puzzle.

Rule # 1: Cross Referencing in the rows.

Rule # 2: Cross Referencing in the columns.

Rule # 3: Cross Referencing in the box. Only one number can filled that particular cell in the box when we look around that cell in related rows, related columns and related box. We have to remember that there are nine boxes in the grid. From left to right and up to down, cells a1, a2, a3, b1, b2, b3, c1, c2, c3 constitute box # 1; cells d1, d2, d3, e1, e2, e3, f1, f2, f3 constitute box # 2; and so on until the cells in box # 9 comprise cells g7,g8, g9, h7,h8, h9, i7, i8, i9.

Rule # 4: Three or less remaining numbers in a box.

Rule # 5: Three or less remaining numbers in a row.

Rule # 6: Three or less remaining numbers in a column.

The First Frame of the first nine steps towards a solution is shown below:

Step #	Rule #	Look At	Observe That	Cell #	Equals
1	1	row 1-2-3	d2 = h3 = 7	b1	7
2	1	row 7-8-9	a9 = d7 = 2 = i1	g8	2
3	2	col a-b-c	b2 = a5 = 8 = d9	c7	8
4	2	col g-h-i	g6 = i9 = 4	h2	4
5	1	row 7-8-9	c7 = d9 = 8	i 8	8
6	1	row 1-2-3	b2 = f1 = 8 =i8	g3	8
7	3	cell f6 in box # 5	f3=1; f8=3; g6=4; f5=5; d6=6; f4=7; f1=8; and d5=9.	f6	2
8	6	col f, remaining 4,6,7 :  h2,i9=4	3 remaining no. in 3 remaining cells	f7	4
9	3	cell d8 in box # 8	h8=1; g8=2; f8=3; f7=4; d6=6; d2=7; d9=8; and d5=9.	d8	5

Rules # 7 to # 9: Paring two numbers to two cells in a row (#7), or a column (#8), or a box (#9).

Frame # 2: Steps # 10 to # 18.

Step #	Rule #	Look At	Observe That	Cell #	Equals
10	6	col d, remaining 1,3,4 f3 = 1	f3=1 and d1, d3 cannot be 1	d4	1
11	6	col d remaining 3,4	a1=4, so, d3 = 4	d3	4
12	6	col d last cell	c1 = 5	d1	3
13	1	row 1-2-3	c1 = d2 = 5	e3	5
14	4	box 2 remaining 2,6.9	i1 = f6 = 2	e2	2
15	8	row 1 pairing 6, 9	e1=h1=6,9	g1	1
16	4	box 3 remaining 3,6,9	i5 = 6	h1	6
17	5	row 1 last cell		e1	9
18	4	box 2 last cell		f2	6

Frame # 3: Steps # 19 to # 27.

Step #	Rule #	Look At	Observe That	Cell #	Equals
19	6	col f last cell		f9	9
20	1	row 7-8-9	f7 = i9 = a1 = 4	b8	4
21	2	col g-h-i	g1 = h8 = d4 = 1	i6	1
22	3	cell h9 in box # 9	h8=1;g8=2;g9=3;i9=4;h1=6;h3=7;i8=8;f9=9	h9	5
23	1	row 7-8-9	h9 = d8 = 5	a7	5
24	9	col i, pair 3, 9	i2 = i3 = 39 f4 = 7	i7	7
25	9	col i, pair 3, 9	i2 = i3 = 39  f4 = 7	i4	5
26	1	row 4-5-6	i4 = f5 = a7 = c1 = 5	b6	5
27	8	row 7, pair 6, 9	6,9 in g7 h7	e7	1

Frame # 4: Steps # 28 to # 36.

Step #	Rule #	Look At	Observe That	Cell #	Equals
28	2	col g-h-i	h1 = i5 = 6	g7	6
29	5	row 7 last cell		h7	9
30	2	col a-b-c	a1 = b8 = g6 = 4	c5	4
31	1	row 4-5-6	c5 = g6 = 4	e4	4
32	2	col d-e-f	f1 = d9 = 8 = a5	e6	8
33	4	box 5 last cell		e5	3
34	1	row 4-5-6	a5 = e6 = 8 = g3	h4	8
35	1	row 4-5-6	d4 = i6 = 1	b5	1
36	3	cell g4 in box #6	g1-1;g8=2;g9=3;g6=4;g2=5;g7=6; f4=7; g3=8	g4	9

Frame # 5: Steps # 37 to # 45.

Step #	Rule #	Look At	Observe That	Cell #	Equals
37	1	row 4-5-6	g4 = d5 = 9= c8	a6	9
38	6	col g last cell		g5	7
39	1	row 4-5-6	f4 = g5 = 7	c6	7
40	5	row 6 last cell		h6	3
41	6	col h last cell		h5	2
42	1	row 4-5-6	h5 = f6 = 2 = a9	b4	2
43	5	row 4 last cell		a4	3
44	2	col a-b-c	a9 = b4 = 2 = e2	c3	2
45	2	col a-b-c	a6 = c8 = 9	b3	9

Frame # 6: Steps # 46 to # 55.

Step #	Rule #	Look At	Observe That	Cell #	Equals
46	6	col b last cell		b9	6
47	2	col d-e-f	f2 = d6 = 6 = b9	e8	6
48	2	col e last cell		e9	7
49	1	row 8 last cell		a8	7
50	4	box 7 last cell		c9	1
51	2	col c last cell		c2	3
52	1	row 1-2-3	d1 = c2 = 3	i3	3
53	4	box 3 last cell		i2	9
54	1	row 2 last cell		a2	1
55	2	col a last cell		a3	6

FRANCIS’ RULES FOR SUDOKU:

RULE #1  X-Ref. Horizontal Rows: 1-2-3 or 4-5-6 or 7-8-9

RULE #2  X-Ref. Vertical Cols: A-B-C or D-E-F or G-H-I

RULE #3  X-Ref. Look at each cell - check for rows; cols; box

RULE #4  Every Box: Remaining Cells (3 Cells or less)

RULE #5  Every Row: Remaining Cells (3 Cells or less)

RULE #6  Every Col: Remaining Cells (3 Cells or less)

RULE #7  Boxes: Pairing 2 Cells with the same 2 digits

RULE #8  Rows: Pairing 2 Cells with the same 2 digits

RULE #9  Cols: Pairing 2 Cells with the same 2 digits

RULE #10  Box: Only this digit fits into this Cell

RULE #11  Row: Only this digit fits into this Cell

RULE #12  Col: Only this digit fits into this Cell

RULE #13  Boxes: Trebling 3 Cells with the same 3 digits

RULE #14  Rows: Trebling 3 Cells with the same 3 digits

RULE #15  Cols: Trebling 3 Cells with the same 3 digitsRULE

Now can you try to do the next puzzle below:

Remember, do not be angry with yourself if you are stuck at any time during the solution process. Stay away for a while and come back in an hour or a day’s time and you can suddenly see another fresh angle to tackle the puzzle.

If you are still stuck without a solution, it may well be that the puzzle has no solution to it at all.

Do not border to study how to do difficult Sudokus as such difficult sudokus are meant for those who are mighty good in mathematical problems and we are no mathematical experts.

God Bless all of you.