Sudoku Square and Space Notation

by Ed Glysson
(Morgan Hill, CA.)

Sudoku Square and Space Notation

As we know, a Sudoku Puzzle Template is a 9x9 matrix of 81 Spaces, which are initially all empty.

These 81 Spaces are grouped into 9 Squares; each containing 9 Spaces.

The Spaces within the Squares can be numbered 1 through 9; with the top three Spaces referred to as 1, 2, and 3; the middle Spaces referred to as 4, 5, and 6 and the bottom Spaces referred to as 7, 8, and 9.

The 9 Squares are stacked three high and three wide. The top three Squares can be referred to as Squares 1, 2, and 3. The middle Squares as Squares 4, 5, and 6. And the bottom Squares as Squares 7, 8 and 9.

Instead of referring to a Space location by Row and Column, Spaces can be referred to by Square and Space number. For example instead of referring to Row5:Column5 (R5C5), one can refer to Square5:Space5 (55 for short, the Square is assumed to be the first number and the Space the second).

So, visualize a Sudoku puzzle and the location of Space 11 (the first number is the Square, the second number is the Space). Where is Space 11? That's right, the upper left Space in Square 1. Where is 99? That's right the lower right Space in Square 9. How about 62, 85, 19, 53, and 47? Try not to look at a puzzle and just visualize the locations. Also, use an existing Sudoku puzzle and just point to Spaces at random and identify the Square and Space. When you get the hang of it, it's pretty neat.

Groups of three Squares can be referred to as “Belts” (horizontal grouping of three Squares) and “Curtains” (vertical grouping of three Squares). The Belts are numbered 1, 2, and 3 from Top to Bottom; or can be referred to as Top, Middle, and Bottom, The Curtains are numbered 1, 2, and 3 from Left to Right; or can be referred to as Left, Middle, and Right. In new communications one can refer to the Middle Belt or Right Curtain and the reader can easily visualize that portion of the puzzle under discussion.

So why bother with another method of Space reference?

Well there are a couple of benefits... One is to break away from the spreadsheet paradigm. In a spreadsheet paradigm, a Cell containing a number is usually involved in a math operation (add, subtract, etc.). For those familiar with spreadsheets, referring to Rows, Columns, and Cells in a Sudoku puzzle is just teasing a part of your brain that wants to employ spreadsheet type of thinking.

The “numbers” used in a Sudoku game are not numbers. They have no value. An “8” is not “higher” than a “3”. There is no subtraction or addition. The “numbers” used are simply nine different symbols. Try substituting some Alphabet characters for some “numbers”.

I substituted an “E” for a “2” and a “G” for an “8” (my initials EG for my favorite number 28, see example attachment). The puzzle doesn't become unsolvable. It is otherwise the same as the original. (Admittedly using numbers as symbols is still handy for checking your work so I don't recommend changing to other symbols as a regular habit.)



The positive difference in recognizing the puzzle entries as symbols instead of “numbers” is that the puzzler begins to think more in terms of pattern recognition rather than mathematics. Acknowledging that Sudoku uses symbols instead of numbers promotes non-numerical thinking. It's more about patterns and nothing to do with arithmetic. Trying to minimize the use of spreadsheet terms will help to stay focused on pattern searches.

Yet I think the main benefit of referring to Squares and Spaces is ease of pinpointing a Space location. If I suggest that you examine the possibilities for Space 65; it's easy to visualize the specific Space that I am referring to; It's Square 6 Space 5. And if I say compare 45 and 65 for a Hidden Pair. It's very easy to identify the locations in your mind and on the puzzle. But if I say “Compare the Spaces Row5:Column 2 with Row5:Column 8”, most puzzlers will have to refer to the puzzle and count down 5 Rows and then over to Column 2 and then over to Column 8. The puzzler will usually not be able to visualize the referenced locations in their heads.

Also using something I refer to as “Square and Space notation” makes it much easier to document observations. Here is a puzzle advancement written with Square and Space notation (see example attachment):



This reads: Square 7 Space 1 cannot equal a 6; because if Square 7 Space 1 equals a 6, then Square 8 Space 6 equals a 6, and Square 4 Space 6 equals a 6, leaving no possibility of Square 5 to contain a 6. OK... I admit it's kind of wordy if one has to talk it out, but when you get used to Square and Space reference numbers you don't have to translate them into words. It's just 71, 86, 46, etc. If you are currently making notations with Rows and Columns, try the Square and Space method, I'm sure you will appreciate it.

(For my personal reference, I make copies of puzzle challenges and note their solutions with these notations. Then later I can return to a puzzle challenge and review the solution previously found. Also one can categorize the solutions based on the type of observation that allowed the puzzle to advance. This activity produces a library of puzzle challenges and responses (observations) which is entertaining and beneficial to occasionally review.)

So, hopefully, this article nudged you forward in perceiving a Sudoku puzzle as a set of symbols and not numbers so that your puzzling attentions will be more inclined towards pattern recognition. Also, by using Squares and Spaces to refer to locations in your puzzle, you can more clearly document and communicate your puzzle solutions and observations.


Ed Glysson, Morgan Hill, CA Summer, 2014

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