Minimum number of different clues in a Sudoku

Question

I wonder if there are proper $9\times9$ Sudokus having $7$ or less different clues. I know that $17$ is the minimum number of clues. In most Sudokus there are $1$ to $4$ clues of every number. Sometimes I found a Sudoku with only $8$ different clues.

In this example the number $9$ is missing, but the Sudoku was very well solvable. Is it possible to have a $9\times9$ Sudoku with less than $8$ different clues?

Answer 1

If I understand you correctly the answer is no. If the only numbers in the initial grid are $1,2,3,4,5,6,7$ then in any solution you will be able to swap $8$ and $9$ and you will still have a valid solution.

Answer 2

Does the mere interchangeability of 2 or more missing values really result in more than one solution?

In a standard sudoku these 'values' are only symbols and bear no arithmetical value or meaning. Follow the reasoning above and not any standard sudoku with even 8 different clues can have a unique solution, because you are free to fill in the 9 free cells with any individual number (or icon of clover, pet animal or ladybug).

I think, that as long

1. as the 8, 7 or less given symbols allow only one result among them (and with enough givens they will) and if
2. the remaining available cells can be filled in only one way to accommodate any 1, 2 or more different arbitrary symbols

that ought to be regarded as a unique solution and thus, the puzzle in question as well-posed. I am not entirely sure if this is possible to construct, but think it's rather likely. Thank you for taking up this interesting question.

edit: As an example, I took the unique solution to a trivial standard sudoku, removed 2 of the symbols and searched for the ways to share the free cells.

A puzzle with only 7 different values given:

-------------------------
| 1 4 7 | 2   8 | 3 6   |
| 2   8 | 3 6   | 4 7 1 |
| 3 6   | 4 7 1 |   8 2 |
-------------------------
| 4 7 1 |   8 2 | 6   3 |
|   8 2 | 6   3 | 7 1 4 |
| 6   3 | 7 1 4 | 8 2   |
-------------------------
| 7 1 4 | 8 2   |   3 6 |
| 8 2   |   3 6 | 1 4 7 |
|   3 6 | 1 4 7 | 2   8 |
-------------------------

And the only possible distribution of the missing values (named arbitrarily):

-------------------------    -------------------------
|       |       |     A |    |       |   B   |       |
|       |     A |       |    |   B   |       |       |
|     A |       |       |    |       |       | B     |
-------------------------    -------------------------
|       |       |   A   |    |       | B     |       |
|       |   A   |       |    | B     |       |       |
|   A   |       |       |    |       |       |     B |
-------------------------    -------------------------
|       |       | A     |    |       |     B |       |
|       | A     |       |    |     B |       |       |
| A     |       |       |    |       |       |   B   |
-------------------------    -------------------------

Every symbol in a standard sudoku's solution is placed according to one out of 46656 distribution patterns.

My understanding of a 'unique solution' could be expressed like: There exists exactly one set of 9 patterns that can be combined without overlapping and where each pattern covers one of the (given or missing) symbols.

Returning to the OP's question, my answer would be 'yes', sudoku puzzles may have less than 8 different values given and still result in a unique solution.

Which brings up some other questions to be maybe explored in the future:

1. How much further can the number of different values be decreased?
2. How much could an individual value's count be decreased?
3. And how are these parameters related?

I really wouldn't know how to find proof in any form.