Comments on: How many Sudokus?

Comment on How many Sudokus? by: Chad Musick

Mon, 21 Jul 2008 08:32:34 +0000

I think the point Barry is making is that any solution which has at least 73 clues is guaranteed by the pigeonhole principle to have a full group of 9 filled in. Removing one of these 9 leaves no new ambiguity; if it wasn’t ambiguous before, it isn’t after. The bound would be when the removal of any digit, not just a particular one, makes the puzzle ambiguous.

Comment on How many Sudokus? by: Kevin Cormier

Sun, 22 Jun 2008 23:17:44 +0000

@Barry - If two of the missing 4 digits exist in the same block, and the other two are in rows/columns so that they form a square, then the puzzle has multiple solutions. Brian’s answer of 78 is in fact correct as far as I can tell.

Comment on How many Sudokus? by: Mary L. Brown-Wallace

Tue, 08 Apr 2008 15:17:35 +0000

even if solutions to sudoku puzzles are found and mastered today, tomorrow the stragety of the sudoku puzzle creators will change just to keep us on our toes. Thanks for your solutions.

Comment on How many Sudokus? by: George Bell

Wed, 19 Mar 2008 23:47:54 +0000

It seems to me Sudoku puzzle creators are always careful that their puzzles have unique solutions, but it is not clear to me if they make sure that removing any clue results in multiple solutions. Is this really the case? In any case, the question Barry raises is interesting. Can one get a lower bound just by looking at a bunch of Sudoku puzzles and taking the largest number of clues? I doubt it for the reason given in my first sentence.

A much simpler hexagonal-grid variant goes by the name of Septoku. I have analyzed this puzzle, and there are only 6 possible solutions, and the minimum number of clues is 6. I’m not sure what the total number of possible puzzles or maximum number of clues, but these numbers should be very easy to compute in this case. You can find out more about this puzzle by googling “Septoku”.

Comment on How many Sudokus? by: Barry Cipra

Sun, 02 Mar 2008 01:22:26 +0000

@Brian: The obvious, trivial upper bound on the number of clues in an irreducible puzzles is 72, not 78. No row (or column or 3×3 subgrid) needs more than 8 clues.

@Claire: This is great. I look forward to seeing what the modeling contestants come up with.

Comment on How many Sudokus? by: Claire

Sat, 01 Mar 2008 17:44:30 +0000

Problem B in this year’s Mathematical Contest in Modeling was to devise an algorithm for generating Sudoku puzzles of various difficulty levels. The submissions are usually evaluated in March, so maybe we’ll see some interesting new approaches (and maybe some of the teams will turn their work into actual software!).

Comment on How many Sudokus? by: brian

Sat, 01 Mar 2008 02:22:43 +0000

@ Barry: Seems like a hard problem, but I can solve the very easiest trivial bit: The maximum maximum is 78. The reason is that no grid with only one, two or three blanks can be ambiguous: It has either one solution or none. Thus, starting with a fully filled-in valid solution, you can always remove three entries to leave a 78-clue puzzle. But four blanks *can* be ambiguous. Suppose they form a square block spanning two rows, two columns, and two blocks and can be filled in:

12
21

Then they will also accept the entries:

21
12

So we have to supply either a 1 or a 2 somewhere in the block as a clue in order to suppress the ambiguity. Of course it might still be possible to remove some other number elsewhere, so this is only an upper bound on the maximum. Not even a very useful one — but I worked this out while negotiating stop-and-go traffic on the Deegan Expressway through the Bronx, an it’s the best I have to offer.

Comment on How many Sudokus? by: Barry Cipra

Thu, 28 Feb 2008 23:48:33 +0000

Sudoku theorists are rightly fixated on finding the minimum number of clues necessary for a puzzle to have a unique solution, but I’d like to raise a question in the opposite direction: What is the *largest* possible “irreducible” set of clues? By “irreducible” I mean that the removal of any clue produces a puzzle with more than one solution. In general, each of the (roughly) 5.5 billion Sudoku solutions comes equipped with a collection of irreducible sets of clues. These irreducible sets of clues have various sizes. Among all them, there will be one (or more) with the least possible number of clues and one (or more) with the most. The former number is thought to be 17; I’m wondering about the latter.