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The standard rules for sudoku say that you have a 9×9 grid and need to put in every digit from 1 to 9 in a way that each digit occurs exactly once in each row, column and 3×3 box.

So the grid can be separated into 9 distinct groups, where each group only has one digit. These groups have one cell in each row, column and box and are completely disjoint.

But did you know that that is not true when looking at subgroups of cells?

Find a group of cells, such that:

  • There are exactly two cells in each row, column and 3×3 box
  • The group can NOT be separated in two groups of exactly one cell in each row, column and 3×3 box.
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1 Answer 1

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The following set of cells meets the requirements:

Blank sudoku grid with 18 cells highlighted

Proof:

If the top-left cell (labelled "1") is in one group (the orange group). Then the cells in the same row/column/3x3 (labelled "2") must be in the other group (the green group).
Then the cells in the same row/column/3x3 as those (labelled "3") must be in the orange group, and then the "4"s must be in the "green" group.
Continuing in the same way, we end up with two "5"s in the same row that must be orange showing that it is impossible to split these eighteen cells into the two sets of nine.
enter image description here

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  • $\begingroup$ Nice! Can you find a formula for the construction? So if I give you a set and you tell me if its valid? $\endgroup$
    – Nurator
    Commented Nov 14, 2022 at 22:14
  • $\begingroup$ Simply look for rot13(plpyrf bs bqq yratgu). $\endgroup$
    – AxiomaticSystem
    Commented Nov 15, 2022 at 2:13

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