## NCMS: Non-Consecutive (3x3) Magic Square Sudoku

For fans of Killer Sudoku, Samurai Sudoku and other variants

### NCMS: Non-Consecutive (3x3) Magic Square Sudoku

This sounds contradictory - the 3x3 magic square itself has consecutive cells, so how can one made NC puzzles out of it? But if we spread the cells of the 3x3 magic square then anything is possible - in fact we can have very "restrained" puzzles (as coined by Ruud & elaborated by HATMAN).

Firstly, one of the most orthodox way to inscribe a magic square is to place it as the "centre dots" of the puzzle (i.e. r258c258). Then with the NC property and DG (Disjoint Group) there is always a unique solution for every way you fill the 3x3 magic square. Therefore the following puzzle has a total of 8 solutions:
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`DG NC MS (Disjoint Group Non-Consecutive Magic Square) Sudoku . . . | . . . | . . . . # . | . # . | . # . . * . | . . . | . . .-------+-------+------- . . . | . . . | . . . . # . | . # . | . # . . . . | . . . | . . .-------+-------+------- . . . | . . . | . . . . # . | . # . | . # . . . . | . . . | . . .The # cells hold a 3x3 magic square.Placing each of {12346789} in the * cell gives a unique solution.`

If we "contract" the magic square a bit we can even drop the DG constraint and have a puzzle with only 4 solutions:
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`NC MS (Non-Consecutive Magic Square) Sudoku 1 . # . | # . # | . . . . * . | . . . | . . . . # . | # . # | . . .-------+-------+------- . . . | . . . | . . . . # . | # . # | . . . . . . | . . . | . . .-------+-------+------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . .The # cells hold a 3x3 magic square.Placing each of {1379} in the * cell gives a unique solution.`

Finally, if we're allowed to "tilt" the magic square a bit we can have a puzzle with as few as 2 solutions:
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`NC MS (Non-Consecutive Magic Square) Sudoku 2 . . . | # . . | . . . . . . | * . # | . . . . . # | . . . | . # .-------+-------+------- . . . | . # . | . . . . # . | . . . | # . . . . . | # . . | . . .-------+-------+------- . . . | . . # | . . . . . . | . . . | . . . . . . | . . . | . . .The # cells hold a 3x3 magic square.Placing each of {37} in the * cell gives a unique solution.`

Note that in all 3 cases, we can produce a "one-given" puzzle by placing an odd number under the top-left cell of the magic square (the * cells).

Now with just NC & MS constraints can we get a puzzle with a unique solution? The answer is no, and the proof is left as a challenge for the readers.
udosuk

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Joined: 17 July 2005