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
