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A complete (or solved) sudoku is said to be 3-term arithmetic progression free (3-AP free) if none of its rows or columns contain three terms which are in arithmetic progression (i. e. in none of its rows or columns, for each x and y in them, the number (x+y)/2 does not appear between x and y).

  • a) Exhibit a complete 3-AP free complete sudoku.

  • b) What is the least number of clues required for this particular sudoku to be solvable?

  • c) Among all possible 3-AP free complete sudokus, which requires the least clues for it to be solvable?

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  • $\begingroup$ What if there are no complete 3-AP free sudoku? $\endgroup$
    – Daniel Mathias
    Commented Jan 5 at 3:57
  • $\begingroup$ Does the rule apply only to 3 adjacent cells? $\endgroup$
    – Florian F
    Commented Jan 8 at 16:16
  • $\begingroup$ @FlorianF No, any three cells in the same row or column. But it would be worth looking into the matter when that is indeed the case. $\endgroup$
    – Bernardo Recamán Santos
    Commented Jan 9 at 21:53

1 Answer 1

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All these questions are easy to answer, because...

No 3-AP-free sudoku exists because the special restriction implies 5 cannot go in even-indexed rows or columns (using 1-indexing), whereas there has to be a 5 in all rows and columns by sudoku rules. Since it lies in the middle of arithmetic progressions 159, 258, 357 and 456, 1 and 9 must lie on the same side of 5 in a given line (row or column), and similarly for 28, 37 and 46. Thus there must be an even number of squares before and after the 5.

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