Now let us take one example from NZ Herald’s daily Sudoku - an easy one - from Monday 16th February 2016.

Please refer to the photo above for reference numbers. Rows are numbered 1 right through to 9. Columns are alphabetically referenced a right through to i. Boxes are referenced from Box # 1 at the top right-hand corner going from left to right until Box # 9 with cells g7, g8, g9, h7, h8, h9, and i7, i8, i9. The small handwritten numbers at the bottom right corner of each cell represent the order of the steps in the solution to this challenge.

Let’s start.

Step 1: Francis’ Rule # 1: Looking at rows 1, 2, 3 and Boxes # 1, # 2 and # 3, we find that cell a3 = cell d2 = 4 = cell h6 and = cell i7. So, the cell g1 has to be a 4. All right with this so far?

Steps 2 and 3: Francis’ Rule # 1: Look at rows 4, 5 and six in Boxes # 4, # 5 and # 6. We have cell i4 = cell a5 = 6. So the cell e6 in Box # 5 just be a 6. Similarly, we have cell i5 = cell d6 = 1. Thus cell a4 must be a 1. QED.

Step 4: Francis’ Rule # 1 still: Rows 7, 8 and 9 together with Boxes # 7, # 8 and # 9; cell i7 = cell e8 = 4 = cell a3 = cell c5. Thus cell b9 = 4. Just a bit more areas to look into, of course.

Steps 5 and 6: Francis’ Rule # 2: This time let us look vertically rather than horizontally at columns d, e and f in conjunction with Box # 2, Box # 5 and Box # 8. Cell e2 = cell f7 =2; so cell d5 in Box 5 must be a 2, right? Similarly in columns g, h and i together with Boxes # 3, # 6 and # 9, we have cell i3 = cell g7 = 3; so cell h5 = 3. So far so good?

Steps 7 and 8: Francis’ Rule # 4, remaining cells in Box # 6. There are two cells remaining here, cells g4 and i6. The two missing numbers for this Box # 6 are 7 and 8. Because cell d4 = 7 so cell g4 = 8 and cell i6 = 7, correct?

Steps 9 and 11: Francis’ Rule # 1: Now that cell i6 is found to be = 7 then looking at rows 4, 5 and 6 and Boxes # 4, 5 and 6, it is easy to conclude that cell b5 = 7 (Step 9). Similarly now that cell h5 = 3 and cell b6 is given to be 3 as well, cell e4 = 3 (Step 11).

Step 10 uses Francis’ Rule # 4 for cell c6 = 2 as this is the only remaining cell in Box # 4.

Steps 12, 13 and 14: Francis’ Rule # 5: Look at the remaining three cells for row 3. These numbers are 2, 7 and 8. Since cell e2 = 2, cells e3 and f3 from the same Box # 2 cannot have a 2. Thus cell b3 = 2 (Step 12). The cell f8 = 7, so cell e3 = 7 (Step 11) and cell f3 = 8.

Steps 15 and 16: Francis’ Rule # 5: Similarly if we look at row 5, we can conclude that cell e5 = 8 (Step 15) and cell f5 = 5 (Step 16).

Step 17: Francis’ Rule # 2: Looking at columns a, b and c; cell a9 = 2 as cell b3 = cell c6 = 2.

Steps 18, 19 and 20: back to Francis’ Rule # 5 at row 7 where the remaining numbers for the three remaining cells are 5, 6 and 8. With cell d3 = 5 and cell d9 = 8, cell d6 = 6 (Step 18). Then with cell 5 = 8, then cell h7 = 8 (Step 19) and cell e7 = 5 (Step 20).

Step 21: Francis’ Rule # 2: columns a, b and c where cell c3 = cell a5 = 6, then cell b8 must = 6.

Step 22: Francis’ Rule # 1: Looking at rows 7, 8 and 9, cell h7 = cell d9 = 8 and so cell e8 = 8.

Step 23 was wrongly tagged as Step 24 in the photo above. Francis’ Rule # 4 applies here where the remaining cell e9 = 5.

Now why not try to see which Francis’ Rules apply to give cell f9 = 3 (Step 25), cell d1 = 3 (Step 26), cell f2 = 1 (Step 27) and cell e1 = 9 (Step 28).

Francis’ Rule # 6 applies to column e to give cell e9 = 1 (Step 29) and Francis’ Rule # 4 applies to box 8 to give d8 = 9.

We can now easily deduce the numbers for the remainder cells.

Step Francis’ Rule # Cell Equals

30 2 cols g, h, i g9 6

31 6 column g g8 2

32 6 column g g2 7

33 1 rows 7, 8, 9 h8 1

34 4 Box # 9 i8 5

35 4 Box # 9 h9 7

36 4 Box # 9 i9 9

37 4 Box # 3 i1 3

38 4 Box # 3 h1 2

39 4 Box # 3 h2 5

40 2 Cols a, b, c a1 5

41 6 Column a a2 9

42 5 Row 2 c2 3

43 5 Row 2 b2 8

44 5 Row 1 b1 1

45 5 Row 1 c1 7