Generalization of Sudoku

Everything about Sudoku that doesn't fit in one of the other sections

Generalization of Sudoku

Postby Esteban_Zia » Tue Nov 15, 2016 12:16 pm

Sudoku generalization

with the following notations :
I block line counter, J block column counter,
i line counter inside a given block, j column counter inside a given block,
the location of each element of a Sudoku grid is defined by four digits, each one belonging to the set {1,2,3}.

For example, the element at the crossing of the third line and the fifth column has the following indexes :
I=1 i=3 J=2 j=2

Thus the Sudoku becomes a quadri-dimensional tensor aI,J,i,j of 3*3*3*3 dimension.

Whith these notations, Sudoku constraints can be rewritten in the following way :

Each element of the grid must b a digit (belonging to the set {1,2,3,...,9}).

Two distinct elements aI,j,i,j et aI',J',i',j' must be different if
(I,i) = (I',i') (elements belonging to the same line)
(J,j) = (J',j') (elements belonging to the same column)
(I,J) = (I',J') (elements belonging to the same block)

This leads us to the definition of a super-sudoku where these equalities would be true for each couple of indexes.

Its definition is quite simple.

« A super-sudoku grid is a 3*3*3*3 tensor, whose elements are digits and whose elements sharing two indexes (but not four !) must be different. »

In other terms, referring to the actual two-dimensional grid of Sudoku, this adds the following constraints :
two elements having the same location inside distinct blocks must be different (for ex, the top left elements of each block),
all the ith lines of the blocks located on the same « vertical » must have different values,
all the jth columns of the blocks located on the same « horizontal » must have different values.

A bit less beautiful but certainly easier to practice, :), would be a Plus-Sudoku adding only the first of these three constraints to the "normal" Sudoku.

Best regards,

Etienne Turpin
Brussel
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Re: Generalization of Sudoku

Postby dobrichev » Wed Nov 16, 2016 6:24 am

Hi Etienne Turpin, and welcome to the forum.

See http://forum.enjoysudoku.com/sudoku-is-a-6-dimensional-problem-t2151.html where the famous Guenter Stertenbrink (dokuso) sees Sudoku as a 6-dimentional problem.
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Re: Generalization of Sudoku

Postby David P Bird » Wed Nov 16, 2016 11:36 am

Code: Select all
33 65 78 | 12 44 87 | 21 56 99
52 19 94 | 61 28 76 | 43 37 85
47 81 26 | 59 93 35 | 68 72 14
------------------------------
69 92 15 | 48 71 24 | 57 83 36
88 46 31 | 97 55 13 | 79 64 22
74 27 53 | 86 39 62 | 95 18 41
------------------------------
96 38 42 | 75 17 51 | 84 29 63
25 73 67 | 34 82 49 | 16 91 58
11 54 89 | 23 66 98 | 32 45 77

Here are two 'super' grids, using the left and right digits. They were derived at the request of a member of a crocheting circle who wanted a Sudoku-based <Afghan Pattern>.

In this case no box-row or box-column is repeated and every left/right pairing occurs just once.

The original discussion is lost but this solution was reported <in this thread>.

DPB
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