7
$\begingroup$

"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.

Gareth McCaughan poses an interesting question:

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?

$\endgroup$
10
$\begingroup$

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 . . . .

check 1 check 2 check 3

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

$\endgroup$
  • $\begingroup$ The difficultly ratings of the three are, from left to right: Moderate, Very Easy, Easy. $\endgroup$ – agc Jun 25 at 12:06
  • $\begingroup$ 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. $\endgroup$ – Jaap Scherphuis Jun 25 at 12:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.