Sudokion

SUDOKION is a family of hand-crafted spatial-logic puzzles derived from Sudoku.
The rules of SUDOKION are identical to those employed by Sudoku: the set of values of the puzzle - usually numbers, but possibly letters, icons or symbols indicating equivalent value - must appear in every row, column and cluster - once and only once.
As SUDOKION employ irregularly-shaped clusters the size of the puzzles may be scaled up or down from Sudoku’s classic nine-by-nine, 81-cell format. The smallest SUDOKION comprise 36 cells while the largest - other than Super SUDOKION (similar but not necessarily identical to Samurai Sudoku) - contain 121 cells.
SUDOKION are produced in Plain and Plus Format. Plain Format is the unadorned puzzle. Plus Format involves the superimposition on the puzzle of an X, V line or diagonal line - in the same way that an X is superimposed in the case of X Sudoku.
All puzzles produced in Plain Format have at least one fragmented cluster. The Hypernion has one cluster broken into two or more parts. One cluster in the Pandemonion is completely fragmented, while the Katastrophion is an amalgam of the Hypernion and the Pandemonion, with one cluster completely fragmented and another broken into two or more parts.
Sudokion Examples
The cluster colours have no particular significance other than to guide the solver’s eye.
A prerequisite of every SUDOKION is that it be both proper and satisfactory, i.e. that it offers a unique solution and that the solver need not resort to trial and error.
The creator’s primary objective, when creating a new puzzle, is economy of clues. Whereas the minimum number of clues for a proper Sudoku is 17 and 12 for a proper X Sudoku, many Plain SUDOKION contain fewer than 17 clues while many Plus SUDOKION require fewer than 12 clues.
The creator’s secondary objective is the production of puzzles with as wide a range of cluster shapes and arrangements as possible.
The creator’s third objective is making the SUDOKION as challenging as possible.
Examples of Plain Sudokion
Hypernion

The Hypernion is produced in 49-, 64- and 81-cell formats.
Rules: The set of the values of the puzzle must appear in every row, column and cluster. The pink cluster is fragmented into two or more parts; it, too, must contain the set of the values.
Set of values in this example: 8
Pandemonion

36-, 49-, 64-, 81- and 121-cell formats.
Rules: The set of the values of the puzzle must appear in every row, column and cluster. The green cluster is fragmented into parts equivalent to the set of the values; it, too, must contain the set of the values.
Set of values in this example: 11
Katastrophion

49-, 64- and 81-cell formats.
Rules: The set of the values of the puzzle must appear in every row, column and cluster. The pink cluster is fragmented into two or more parts while the green cluster is fragmented into parts equivalent to the set of the values; both these must also contain the set of the values.
Set of values in this example: 9
Examples of Plus Sudokion
In the case of the Plus SUDOKION a red line or lines is superimposed on the puzzle. If there is one superimposed line - a diagonal or a V - the set of the values of the puzzle must appear on that line. If there are two superimposed lines - an X or a parallelogram - the set of the values of the puzzle must appear on both lines, in the case of the X on both diagonals, in the case of a parallelogram on both the V and inverted-V lines.
D Logikion

64- and 81-cell formats.
The set of the values of the puzzle must appear in every row, column and cluster. In addition, the set of the values must appear on the red diagonal line.
Set of values in this example: 8
V Pandemonion

81-cell format.
The set of the values of the puzzle must appear in every row, column and cluster. The green cluster is fragmented into parts equivalent to the set of the values; it, too, must contain the set of the values. In addition, the set of the values must appear on the red, V, line.
Set of values in this example: 9
X Katastrophion

81-cell formats.
The set of the values of the puzzle must appear in every row, column and cluster. The pink cluster is fragmented into two or more parts while the green cluster is fragmented into parts equivalent to the set of the values; both these must also contain the set of the values. In addition, the set of the values must appear on both red diagonal lines.
Set of values in this example: 9
Economy of clues
The P (Parallelogram) Pandemonion that appears here is the most economic of all the SUDOKION created by Stephen Jones. While it contains only nine clues, it possesses a unique and satisfactory solution, not requiring the solver to resort to guesswork.
P (Parallelogram) Pandemonion

The set of the values of the puzzle must appear in every row, column and cluster The green cluster is fragmented into parts equivalent to the set of the values; it, too, must contain the set of the values. In addition, the set of the values must appear on both the V and inverted-V lines.
Set of values in this example: 9
Proof of unique solution are provided for examples of SUDOKION with fewer than 12 clues:
11 clues X Pandemonion, V Pandemonion
9 clues P Pandemonion
The reason that SUDOKION may be created with a greater economy of clues than Sudoku is that the negative and positive evidentiary strength of SUDOKION clues is almost always greater than their Sudoku equivalents.
Negative evidentiary strength of Sudokion clues

In this example, with only one clue revealed in the Sudoku - a 9 placed in cell E4 - the value 9 is immediately blocked (without need of any further clues) from a total of 20 other cells in five clusters. So, this clue’s negative evidentiary strength is 20.

In this example, with only one clue revealed in the Katastrophion - a 9 placed in cell E4 - the value 9 is immediately blocked (without need of any further clues) from a total of 52 other cells in all nine clusters. So, this clue’s negative evidentiary strength is 52.
While a single revealed Sudoku clue will always have a negative evidentiary strength of 20, the evidentiary strength of a single revealed SUDOKION clue will vary from case to case. In almost all cases its negative evidentiary strength will be greater than 20.
Positive evidentiary strength of Sudokion clues
Whereas Sudoku requires at least four clues to establish the location of a placement, in many cases only one clue is required in SUDOKION.

In the Sudoku example at right the clues at A2, D3, G1 and H8 are sufficient to prove that 5 must occupy cell I9. The reverse, however, is not true: the clues at D3, G1, I9 and H8 do not establish that 5 must occupy cell A2.

In the Logikion example at right, a 5 placed at A2 forces a 5 into I9. And, of course, a 5 placed at I9 forces a 5 into A2.

Going back to the Katastrophion used to demonstrate negative evidentiary strength a 9 placed in any of the cells E4, C8 or A9 will force the placement of the other two. They also force a 9 into either of cells B6 or B7.
History
Invented and created by Stephen Jones, SUDOKION currently comprise four types - Logikion, Hypernion, Pandemonion and Katastrophion, nominated in order of increasing complexity.
Stephen started making Logikion, also known as Shape Shifters, for his wife, Michele Day, in February 2010, as an alternative to Sudoku which she had recently taken up. By February 2011 Stephen had created the Hypernion, Pandemonion and Katastrophion, all in Plain Format. He created the first of the Plus-Format SUDOKION, an 81-cell Pandemonion on 1 May 2011.
Nowadays, Stephen creates 26 types of SUDOKION, including Super SUDOKION of various types and sizes, in Plain and Plus Format.
First publicity
The first newspaper article on SUDOKION appeared in more than 20 Australian Provincial Newspapers down the east coast of Australia on 28 October 2010.
First publication
The first SUDOKION ever published - a 64-cell Logikion - appeared in the may 2011 edition of Bangalow Heartbeat, a monthly publication in Byron Shire in the north-east of New South Wales, Australia. A new SUDOKION appears in the same publication every month.
First mass exposure
The first mass exposure of the SUDOKION occurred on Adam Spencer’s Breakfast Show on the Australian Broadcasting Corporation’s 702ABC radio based in Sydney, New South Wales, Australia, in early August 2011. An easy and a difficult SUDOKION are published on Mr Spencer’s blog every Friday morning.
Availability to the public
Recently Muddled Puzzles has established a joint venture with Mindnumbers Pty Ltd, a company established to promote SUDOKION and negotiate and implement all hard-copy and electronic publishing arrangements on behalf of Muddled Puzzles.

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