# Design Two Sudoku Puzzles **URL:** \ **Category:** Miscellaneous and Personal Stuff I Must Share\ **Created:** [January 16, 2007, 3:57pm UTC](https://boards.straightdope.com/t/design-two-sudoku-puzzles/388196 "2007-01-16T15:57:15Z")\ **Posts on this page:** 12\ **Page:** 1
**Author:** ![HeyHomie](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/heyhomie/32/207_2.png) [@HeyHomie](https://boards.straightdope.com/u/HeyHomie)\ **Post date:** [January 16, 2007, 3:57pm UTC](https://boards.straightdope.com/t/design-two-sudoku-puzzles/388196/1 "2007-01-16T15:57:15Z")
I’ve recently gotten into Sudoku, and I’ve been wondering about some things. 1. Is there a theoritical minimum number of spaces that can be filled in on a Sudoku puzzle so that the puzzle is solvable? If so, can you point to one such puzzle (or design one yourself)? Conversely… 1. Is there a theoretical maximum number of spaces that can be filled in on a Sudoku puzzle, but yet the puzzle remains unsolvable? Again, if so, can you point to one such puzzle (or design one yourself)? ---
**Author:** ![MissMossie](https://avatars.discourse-cdn.com/v4/letter/m/f6c823/32.png) [@MissMossie](https://boards.straightdope.com/u/MissMossie)\ **Post date:** [January 16, 2007, 9:08pm UTC](https://boards.straightdope.com/t/design-two-sudoku-puzzles/388196/2 "2007-01-16T21:08:16Z")
[QUOTE=HeyHomie] I’ve recently gotten into Sudoku, and I’ve been wondering about some things. 1. Is there a theoritical minimum number of spaces that can be filled in on a Sudoku puzzle so that the puzzle is solvable? If so, can you point to one such puzzle (or design one yourself)? [/QUOTE] Yes, there is. The theoretical minumum for a 9x9 sudoku square is 17 clues. [This site](http://sudokugarden.de/en/info/minimal) has a brief discussion on it and links to such sudokus. [QUOTE=HeyHomie] 2) Is there a theoretical maximum number of spaces that can be filled in on a Sudoku puzzle, but yet the puzzle remains unsolvable? Again, if so, can you point to one such puzzle (or design one yourself)? [/QUOTE] According to this site (warning: pdf), the maximum number of clues for an unsolvable 9x9 sudoku is 33; there is not an example though. ---
**Author:** ![4.66](https://avatars.discourse-cdn.com/v4/letter/4/2bfe46/32.png) [@4.66](https://boards.straightdope.com/u/4.66)\ **Post date:** [January 16, 2007, 10:09pm UTC](https://boards.straightdope.com/t/design-two-sudoku-puzzles/388196/3 "2007-01-16T22:09:50Z")
That’s quite a confusing definition of _solvable_ you’re using there. I’d say a sudoko puzzle is solvable if a solution exists. That is, if all the open spaces can be filled in in such a way that there are no contradictions. So the answer to question 1 would be zero (the solution of any sudoko would be a solution) and 81 would be the other answer. What you’re questioning here is the _uniqueness_ of solutions. Not that this answer is of any help… ---
**Author:** ![Sage\_Rat](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/sage_rat/32/399_2.png) [@Sage\_Rat](https://boards.straightdope.com/u/Sage_Rat)\ **Post date:** [January 17, 2007, 4:05am UTC](https://boards.straightdope.com/t/design-two-sudoku-puzzles/388196/4 "2007-01-17T04:05:41Z")
[QUOTE=4.66] That’s quite a confusing definition of _solvable_ you’re using there. I’d say a sudoko puzzle is solvable if a solution exists. That is, if all the open spaces can be filled in in such a way that there are no contradictions. [/QUOTE] The rule of Sudoku is that there is always a unique solution. So by definition, a Sudoku puzzle which doesn’t have a unique solution is unsolvable. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [January 17, 2007, 4:38am UTC](https://boards.straightdope.com/t/design-two-sudoku-puzzles/388196/5 "2007-01-17T04:38:34Z")
[QUOTE=MissMossie] Yes, there is. The theoretical minumum for a 9x9 sudoku square is 17 clues. [This site](http://sudokugarden.de/en/info/minimal) has a brief discussion on it and links to such sudokus. [/QUOTE] As your link mentions, that result is more empirical than theoretical. As for the maximum, I think I’ve seen a grid with only four empty spaces and two solutions, but I can’t find a reference right now. ---
**Author:** ![j\_sum1](https://avatars.discourse-cdn.com/v4/letter/j/8baadc/32.png) [@j\_sum1](https://boards.straightdope.com/u/j_sum1)\ **Post date:** [January 17, 2007, 6:58am UTC](https://boards.straightdope.com/t/design-two-sudoku-puzzles/388196/6 "2007-01-17T06:58:00Z")
Four empty spaces and two possible solutions is relatively common – I have found a few of these in newspapers and such – you can use logic and deduction to complete everything except the final four squares. I will see if I can find one. ---
**Author:** ![j\_sum1](https://avatars.discourse-cdn.com/v4/letter/j/8baadc/32.png) [@j\_sum1](https://boards.straightdope.com/u/j_sum1)\ **Post date:** [January 17, 2007, 7:30am UTC](https://boards.straightdope.com/t/design-two-sudoku-puzzles/388196/7 "2007-01-17T07:30:21Z")
```auto 3 456 789 654 987 321 789 123 456 5 678 934 436 295 178 897 314 562 541 762 893 972 831 645 368 549 217 ``` The above puzzle does not have a unique solution. You can fill the four blanks with 12/21 or 21/12. As for the answer to the OP… well, I don’t know. But I think it is sensible to distinguish between legal solutions and unique solutions. A well designed puzzle will have a unique solution, but I think the OP was talking about legal solutions. ---
**Author:** ![4.66](https://avatars.discourse-cdn.com/v4/letter/4/2bfe46/32.png) [@4.66](https://boards.straightdope.com/u/4.66)\ **Post date:** [January 17, 2007, 12:39pm UTC](https://boards.straightdope.com/t/design-two-sudoku-puzzles/388196/8 "2007-01-17T12:39:01Z")
[QUOTE=Sage Rat] The rule of Sudoku is that there is always a unique solution. So by definition, a Sudoku puzzle which doesn’t have a unique solution is unsolvable. [/QUOTE] Another rule of sudoku is that the clues don’t contradict each other. Combining our two rules, a sudoku puzzle is, by definition, solvable. But using the word (un)solvable in this way doesn’t make much sense in the context of the OP. [QUOTE=j\_sum1] A well designed puzzle will have a unique solution, but I think the OP was talking about legal solutions. [/QUOTE] Her second question makes clear that she was talking about unique solutions. If she were talking about the other way in which the puzzles can be unsolvable (inconsistent clues: no legal solutions), the question doesn’t make any sense. Once you have such a sudoko you can just keep on adding clues and it remains inconsistent, so it would be a bit strange to ask for a maximum. I think she meant to ask: _What’s the maximum number of clues in a consistent sudoku, such that by adding a clue the sudoku would_ become _either inconsistent or uniquely solvable?_ Your example, combined with the fact that a consistent sudoku with three open spaces has a unique solution, makes clear that the answer is 77. ---
**Author:** ![HeyHomie](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/heyhomie/32/207_2.png) [@HeyHomie](https://boards.straightdope.com/u/HeyHomie)\ **Post date:** [January 18, 2007, 9:54pm UTC](https://boards.straightdope.com/t/design-two-sudoku-puzzles/388196/9 "2007-01-18T21:54:50Z")
[QUOTE=4.66] Her second question makes clear that she was talking about unique solutions. [/QUOTE] I’m a he. ---
**Author:** ![Ludovic](https://avatars.discourse-cdn.com/v4/letter/l/7ab992/32.png) [@Ludovic](https://boards.straightdope.com/u/Ludovic)\ **Post date:** [January 18, 2007, 10:47pm UTC](https://boards.straightdope.com/t/design-two-sudoku-puzzles/388196/10 "2007-01-18T22:47:09Z")
The first rule of Sudoku is: DON’T TALK ABOUT SUDOKU! ---
**Author:** ![cooky173](https://avatars.discourse-cdn.com/v4/letter/c/71c47a/32.png) [@cooky173](https://boards.straightdope.com/u/cooky173)\ **Post date:** [January 19, 2007, 12:35pm UTC](https://boards.straightdope.com/t/design-two-sudoku-puzzles/388196/11 "2007-01-19T12:35:22Z")
Do you mean solvable as in, once x number of squares have been filled in you can use logic to solve the puzzle without resorting to guesses? If that’s what you meant I guess there would be a minimum, and going by **j\_sum1** ’s post it would be 3 blank squares. ---
**Author:** ![HeyHomie](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/heyhomie/32/207_2.png) [@HeyHomie](https://boards.straightdope.com/u/HeyHomie)\ **Post date:** [January 19, 2007, 1:24pm UTC](https://boards.straightdope.com/t/design-two-sudoku-puzzles/388196/12 "2007-01-19T13:24:02Z")
[QUOTE=cooky173] Do you mean solvable as in, once x number of squares have been filled in you can use logic to solve the puzzle without resorting to guesses? If that’s what you meant I guess there would be a minimum, and going by **j\_sum1** ’s post it would be 3 blank squares. [/QUOTE] Yes, that’s exactly what I meant. And that it would have a unique solution.