by **Smythe Dakota** » Tue Sep 23, 2008 2:44 am

It has been pointed out that vanilla sudoku can be thought of as 6-dimensional, with just 3 values allowed in each dimension.

A completed (or incomplete) sudoku grid can be thought of as a set S of sextuples of the form (a,b,c,d,e,f), where each of the six must have a value of 1, 2, or 3, and which is subject to various constraints.

To translate from vanilla to this picture, any time the digit d (1-9) appears in row r, column c, the following sextuple appears in the set S:

(rH,rL,cH,cL,dH,dL)

where r determines rH and rL as follows:

1 --> 1,1

2 --> 1,2

3 --> 1,3

4 --> 2,1

5 --> 2,2

6 --> 2,3

7 --> 3,1

8 --> 3,2

9 --> 3,3

-- and similarly for columns c and digits d.

Constraints would include:

A. No two elements of the set S may contain the same first four coordinates (i.e. any two elements of S must differ in at least one of the first four coordinates). Translation to vanilla: No cell may contain more than one digit.

B. No two elements of the set S may contain the same 1st, 2nd, 5th, and 6th coordinates. Translation to vanilla: No digit may appear more than once in the same row.

C. No two elements of the set S may contain the same 3rd, 4th, 5th, and 6th coordinates. Translation to vanilla: No digit may appear more than once in the same column.

D. No two elements of the set S may contain the same 1st, 3rd, 5th, and 6th coordinates. Translation to vanilla: No digit may appear more than once in the same box.

A completed grid would correspond to a set S (satisfying the above constraints and) having 81 elements. An incomplete grid would correspond to S having fewer than 81 elements.

Bill Smythe