by Ed Glysson
(Morgan Hill, CA)

Sudoku Puzzle — Shuffle 288

Introduction:

This article describes and demonstrates how the entries in a single Sudoku puzzle can be shuffled to create 288 functionally identical puzzles.

Observing the Belt and Curtain layout of a Puzzle:

Let’s start by reviewing the layout of the Puzzle in the context of Belts and Curtains. Examine the attached “Sudoku Puzzle — Triplets with Combinations of Belts and Curtains”. For our purposes we will refer to the Belts as A, B, or C (horizontally from top to bottom), and the Curtains as D, E, and F (vertically from left to right).

(Refer to my other article “Sudoku Square and Space Notation” for a more detail description on how I reference Belts, Curtains, Squares, and Spaces within a Sudoku puzzle.)

The Orientation of the starting puzzle can be viewed as North, East, South, and West (NESW, clock-wise).

Also notice that the attached Sudoku puzzle contains three Triplets which may be used to assist in validating the modified puzzles that will be created via this article. (Refer to my other article “Sudoku Puzzle Triplets – Identifying and Understanding” for further clarifications on Triplets.)

Shuffle Belts and Curtains to Reconfigure the Puzzle:

Now that we have our labeling established we can start to reconfigure the puzzle.

Let’s start by shuffling the Belts into a different order. Instead of the Belts being in ABC order let’s switch them to ACB by moving Belt B below Belt C. The puzzle is still valid; as all of the Squares (and Spaces) have maintained their relationships to both their original Belt and Curtain. By manipulating the Belts we can determine that there are six configurations of Belts:

ABC ACB BAC BCA CAB CBA

Now for each configuration of Belts, the Curtains can be shuffled. Starting with Curtains DEF we have six configurations of Curtains:

DEF DFE EDF EFD FED FDE

So for each of the six configurations of Belts we have six configurations of Curtains for a total of 36 possible combinations. See attachment “Sudoku Puzzle — Triplets with Combinations of Belts and Curtains” for listing of combinations.

Orientating a Puzzle to create additional Perspectives:

Rotating a puzzle clockwise we can re-orient it from the original NESW to WNES, then to SWNE, then lastly to ESWN. We start with the NESW orientation with the top of the puzzle N (at 12 o’clock), then the orientation to the right E (at 3 o’clock), then the bottom orientation S (at 6 o’clock), and lastly the orientation at the left W (at 9 o’clock). As the puzzle is rotated clock-wise the new orientation is identified via top, right, bottom, left.

Observe that we can rotate any of the 36 configured puzzles in four different Orientations and still maintain a valid puzzle. Also, notice that the identity of Belts and Curtains switch with the change in each Orientation. (i.e. Belts become Curtains and Curtains become Belts.)

So, from one valid Sudoku puzzle, with a possible 36 configurations of Belts and Curtains and four Orientations, we can create a total of 144 variations of the original Sudoku puzzle.

Mirroring the Puzzle:

Further we can create an additional 144 variations by taking the mirror image of each of the first 144 variations. Therefore 288 variations of the original puzzle are possible. (Hold a printout of one or more of the puzzles in front of a mirror to observe its mirror image. The mirror image is unique to each puzzle.)

Resolving Puzzle entries:

So how do all of the 288 variations compare when actually doing the Puzzles?

What is really interesting is that all of the 288 variations follow the same process of resolving the puzzle. If the puzzler observes that one of the Spaces in one puzzle can be filled with a “5” then all of the other 287 variations will have a “5” in the same relative location. Each resolution in any Puzzle is exactly the same for all 288 variations. That’s because it’s the same Puzzle. (Try the attached “Sudoku Puzzle – Mix of Six” which were selected at random from the 288 variations. Resolve the Spaces one at a time and observe how each resolution is reflected in each of the other puzzles. Jump around to each puzzle to find the next entry. All discoveries are reflected in the other puzzles.)

Puzzle resolutions are the same because the relationship of the Puzzle Spaces to each other remain the same regardless of the Belt or Curtain shuffle, the Orientation, or the Mirroring of the Puzzle.

Summary:

Each Sudoku Puzzle is one manifestation within an array of 288 variations. Each manifestation is identical to all others in what information is given and what information needs to be puzzled out. The process of puzzling out the entries will be exactly the same for all variations. That’s because they are all the same puzzle. Just the perspective is changing.

Ed Glysson, Morgan Hill, CA Spring, 2016

Attachments:
Sudoku Puzzle – Triplets with Combinations of Belts and Curtains
Sudoku Shuffle – ABCDEF Belt Shuffle NESW
Sudoku Shuffle – ABCDEF Curtain Shuffle NESW
Sudoku Shuffle – Mix of Six

References:
Article “Sudoku Square and Space Notation”
Article “Sudoku Puzzle Triplets – Identifying and Understanding”