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Comment by Anon_troll | Hacker News Reader

> A more reasonable question would be, how many partially completed sudoku grids are there which have a unique solution. We don't know the answer to that.

If you interpret "unique" to mean that two puzzles that lead to the same solution count as one, the answer would be equal to the number of completed grids. Just remove one number and you get a partially completed grid, and there cannot be more.

The interpretation you mean is probably how many partially completed sudoku grids have only a single solution instead of multiple, which leads to a much more interesting question.

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No, in Sudoku, the constraint of a "unique solution" is that a valid puzzle (i.e. an incomplete grid) must only have one correct way of being filled in without violating the rules. If there are two possible ways of filling the grid without violating the rules, the puzzle does not have a unique solution.

It has nothing to do with whether then solution to one puzzle (starting grid) happens to be the same as the solution to a different puzzle (starting grid).

Then the maximum number of solveable puzzles must be the maximum number of unique grids...since you remove grid entries to make the puzzle and if you remove entries with the constraint that the puzzle remains solveable with a single solution etc etc

Why?

If I have a complete grid, I can remove the top left number, and that's a puzzle. Or, I can remove the bottom right number, and that's another puzzle.

So, one unique grid can be the solution to multiple puzzles.

> Why?

Because we said so earlier in this thread.

>> If you interpret "unique" to mean that two puzzles that lead to the same solution count as one

Yap, and that has been refuted:

> No, in Sudoku, the constraint of a "unique solution" is that a valid puzzle (i.e. an incomplete grid) must only have one correct way of being filled in without violating the rules.

My answer is to the answer to the above comment:

> Then the maximum number of solveable puzzles must be the maximum number of unique grids

Imagine you start with one unique solved grid. There's 81 numbers, so by just removing one entry you arguably have 81 different "puzzles" that all lead to the same unique solution. From all of these you can also remove any second number, and get another 6480 puzzles that will still all have the same unique solution.

Now imagine taking away 61 numbers from the solved puzzle to get a sudoku with 20 starting numbers. Some of those puzzles have to be discarded because they might lead to multiple possible solutions, but you will still be left with millions of possible puzzles that look distinct to a human, and require different strategies to solve, all leading to the same solved grid.

In theory yes, but layouts that require just filling in missing numbers are not puzzles. The layouts with missing numbers become puzzles when some thought must go into entry selection. The question is how many puzzles requiring logical deductions are there for each unique 81*81 grid.

At first approximation, the number of logical deductions just goes up (exponentially?) with the number of missing digits. That's why you see many sudoku books grouping puzzles by "difficulty" by just stating how many digits are given.

Of course humans can make much more interesting puzzles, where you are expected to make a certain string of logical deductions to reach the solution. But simple "machine-made" sudokus seems to sell well enough, so I see no reason to exclude them from the definition.

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