# Maximum number of disjoint clue-sets that determine the same Sudoku solution?

"Do sudoku answers always have a single minimal clue set?" No. Solved Sudoku puzzles may have more than one set of disjoint clue-sets. And those clue-sets can vary in difficulty, so one might be easy and another diabolical, even though their 81-number answers are identical.

I wonder what the maximum number of disjoint clue-sets that determine the same solution is. Obviously no bigger than floor(81/17)=4, given that 17 is the minimum number of clues to make a unique solution; I bet 3 is easy but 4 might be difficult or impossible. (On the handwavy grounds that 81 is nearly 5*17, my guess is that 4 is possible.)

So, is it 2, 3, or 4?

## 1 Answer

To start things off, here are three disjoint clue sets with the same solution.

. . . 1 9 . . . .      . 8 . . . 2 . 4 .      7 . 6 . . . 5 . 3
. 4 . . . 3 . . .      5 . 9 . 6 . 2 . 1      . . . 7 . . . 8 .
. . 1 . . 4 7 . .      3 . . . . . . 6 .      . 2 . 8 5 . . . 9
. . . . . . 6 . 8      . 3 . . 4 7 . 2 .      1 . 5 9 . . . . .
. 9 8 . . . . 7 .      2 . . . 3 5 1 . 4      . . . 6 . . . . .
. . . 2 . . . . .      . 6 7 . 8 . . . 5      4 . . . . 1 9 3 .
. . . . 7 9 . 1 .      . 5 2 4 . . 3 . .      8 . . . . . . . 6
. . 4 . 1 . . 5 .      9 . . . . 6 . . .      . 7 . 3 . . 8 . 2
6 . 3 5 . . 4 . .      . 1 . . . 8 . 9 7      . . . . 2 . . . .


I think that finding four disjoint sets will be hard, since 20-clue sets are somewhat rare.

• The difficultly ratings of the three are, from left to right: Moderate, Very Easy, Easy.
– agc
Jun 25 '19 at 12:06
• I should add that the middle and right sets are probably not minimal. I simply took one (minimal) valid sudoku with 23 clues, and then split the other 81-23=58 clues into two valid sudokus. Jun 25 '19 at 12:14