Hello

**satsuma** and

**Richard =capetriangle**.

Thanks to both of you for your participation with me in my latest sudoku discussion.

**Terminology**
I'm pleased that we seem to have consistency in

**terminology** about the sudoku grid:

•

**cell** — the smallest squares in a sudoku grid; the whole grid comprises 81 cells, in a 9x9 array.

•

**row** — a horizontal line of 9 cells in a sudoku grid; the full sudoku grid comprises 9 rows.

•

**column** — a vertical line of 9 cells in a sudoku grid; the full sudoku grid comprises 9 columns.

•

**cage** — a special 3x3 array of 9 cells in a sudoku grid; the full sudoku grid is partitioned into 9 cages.(*)

(*)

Originally I wanted to use **square**, and later **field**, for the special 3x3 arrays of 9 cells making up the sudoku grid. Eventually, following **satsuma**'s lead, I adopted the term **cage**. Evidently that term is widely used in connection with sudoku puzzles.
**Notation**
However, the labelling of the cells, rows, columns and cages has not been uniformly practised in our posts.

Here is the notation I proposed when we previously discussed sudoku puzzles:

Here's how that notation was proposed:

RogerE wrote: ↑07 Dec 2021 20:59
...

**Opportunity for sudoku**
.

Recording the moves in a "game" of

**sudoku** is a practice which has not become popular and widespread. Typically the only assistance offered to players is the publication of the full configuration. The implication is that players can "go figure" how to arrive at that solution — they are offered no insight into a sequence of moves that would reach the solution.

Many will be familiar with the feeling of frustration that occurs when a mistake becomes evident in the course of an attempted solution to a sudoku, but it's almost impossible to retrace the moves to identify where the error actually occurred.

I suggest that an efficient notation for recording sudoku moves would be a great step forward. It would allow the "solution grid" to be accompanied by a list of moves for reaching that solution, so "players" could learn from working through that sequence (especially if they were not able to complete the solution independently).

Furthermore, recording one's moves provides a way of retracing the moves if a contradiction is encountered, so the source of the error can be identified and understood. In summary, notation recording a sequence of sudoku moves would serve as a learning tool.

**Suggested notation for sudoku**
.

Learning from the precedents of chess notation, and the grid labelling in some recent posts in this thread, I would like to suggest the following notation as an experiment for others to try out.

.

.

**Comments**:

• It is helpful to use

**cells** as the name of the 81 smallest elements of the sudoku grid, and

**squares** as the name of the nine (intermediate sized) 3x3 sets of cells. The horizontal lines of 9 cells are

**rows**, and the vertical lines of 9 cells are

**columns**. The symbols to be inserted in the cells are the

**entries**.

• The lower case letters

*a–j* serve to label the rows, while the upper case letters

*A–J* serve to label the squares. The numbers

*1–9* serve to label the columns, and also the entries. The notation distinguishes between numbers as column labels and numbers as entries — illustrated by the same solution below.

• The use of J rather than I is a standard way of avoiding confusion between the letter I and the numeral 1.

...

/RogerE

By contrast, although

**satsuma** also numbered the columns, he labelled the rows with capital [upper case] letters (and did not assign notation to the cages) in his post

It seems to me that lower case letters for rows, and upper case letters for cages, has a more natural association — lower case for the one-dimensional sets, upper case for the two-dimensional sets.

**Using the notation in practice**
.

Specific cells are "addressed" by their row first, column second — following the standard mathematical practice for matrices, and the geographical practice for points on a map. Thus, the top left corner cell of a sudoku grid is a1, the top right corner cell is a9, the bottom left corner cell is j1, and the bottom right corner cell is j9. It's important not to use capital letters for the rows, because capital letters indicate cages. For example, the middle cage in a sudoku grid is E; row e intersects cage E in the cells e4, e5 and e6; the central cell of the whole sudoku grid is e5.

[Most recently, **satsuma** has been using capital letters for the columns, with numbers for the rows, and addressing cells by their column first, row second. In that notation, the top left corner cell of a sudoku grid is A1, the top right corner cell is J1, the bottom left corner cell is A9, and the bottom right corner cell is J9. That's *not* the same as his notation in the post

When there is no introductory explanation of what notation is being used, it takes time and effort to decipher. The chess world has had several quite distinct notations for recording moves and discussing strategies, and that has required serious players to master more than one notation. However, at least within particular large sectors of the chess world, a single notation is the standard followed in publications and records.]
**Using the notation to follow satsuma's reasoning**
.

For example, consider our recently discussed Grid B:

*Grid B*

Just four cells in Grid B contain the entry 7: they are a5, c9, d6 and h4.

In his latest discussion,

**satsuma** has added to Grid B the possible locations where open cells could contain 7. He identifies the cells

b1, b2, b3

e3, e8

f1, f7

g2, g7

j1, j2, j8

As a check, notice that the letters a, c, d, h are absent, and the numbers 4, 5, 6, 9 are absent.

Equivalently, the five remaining entries 7 must be in rows b, e, f, g, j and in columns 1, 2, 3, 7, 8.

However, why have cells f2 and f3 not been included among the possible locations for 7?

I agree with

**satsuma**'s

**case 1**: if

**f1** contains 7, then 7 must be in

**b3, e8, f1, g7, j2**.

I also agree with

**case 2**: if

**g2** contains 7, then 7 must be in

**b1, e3, f7, g2, j8**.

Thus,

**satsuma**'s discussion (with my two added sites) shows the following possible solutions (monochromatic sets) for cells containing 7 :

**b1**, b2, **b3**

**e3**, **e8**

**f1**, f2, f3, **f7**

**g2**, **g7**

j1, **j2**, **j8**

**Case 1**: If 7 is in any one of f1, f2, f3, then row f —> 7 is not in f7, then column 7 —> 7 is in g7, —> 7 is not in g2.

**Case 2**: If 7 is in g2, then row g —> 7 is not in g7, then cage J —> 7 is in j8, then cage F —> 7 is in f7, —> 7 is not in any one of f1, f2, f3.

**Case 3**: If we are not in Case 1 or Case 2, then row f —> 7 is in f7; row g —> 7 is in g7: contradiction in column 7.

Hence either Case 1 or Case 2

*must* apply (assuming the sudoku puzzle has a solution!), so b2 and j1 do not hold 7.

Note that row g requires the cell g2 to contain a number in the set {6, 7}.

From here,

**satsuma** cleverly observes that entries in the cells b2, c2 and c2 together have to comprise the set {2, 3, 8}. Then 8 cannot be in cell a2, so row a, column 2 and cage A require 6 to be in a2.

Then 6 in a2 —> g2 holds 7.

"The rest of the puzzle can then be solved" says

**satsuma**.

A clever piece of reasoning! The fact that Case 1 and Case 2 both have to be explored, and Case 3 briefly used to show that those first two cases are the only viable possibilities, still demonstrates a branch point in this line of reasoning, but the two branches do not have to be pursued so far that one leads to a contradiction; rather, they combine to reach as point where one of them is shown to be "correct".

Thanks

**satsuma**, very interesting indeed!

/RogerE