So isn't this article missing the obvious?
>Correction: this post was edited on the 6 January to reflect the argument that if an n-clue grid is uniquely solvable then adding a digit to make an n+1-clue grid must also be uniquely solvable. So if there are no uniquely solvable 16-clue grids, there cannot be any grids with fewer clues that are uniquely solvable. Thanks to RealMurph and abooij.
Conversely, if a puzzle with n clues is unsolvable, any puzzle with n-1 clues is similarly unsolvable.
"It's easy to see why. A grid with 7 clues cannot have a unique answer because the two missing digits can always be interchanged in any solution."
let's say these are your 7 clues: (row 1) 123 456 7
the article suggests that IF there is a full solution with 123 456 789 then there is a full solution with 123 456 798
but what guarantees that the second (or first) of these 9 clues don't lead to an "impossible" scenario, so that only one of the two is actually possible to solve?
Maybe I am misunderstanding the article.
Turning to what you say: "Suppose there is a unique 15-clue solution. Then add another clue by adding any number from the unique solution. The solution must still be unique (because it contains the 15 clues), so we now have a unique 16-clue solution, which is impossible."
Can you elaborate on why a unique 16-clue solution is impossible?
Essentially those two remaining hints would be variables that you could then substitute either symbol for yielding 2 potential solutions making it an invalid sudoku puzzle.
Let's say there is a unique solution to 123 456 789 . In that solution, swap every 8 and 9. It's not hard to see that this will be a correct solution of 123 456 798. Therefore, 123 456 7 must have an even number of solutions.
For the second question, about why a unique 16-clue solution is impossible: that's the result mentioned in the article, with the proof that took a year to calculate.
In other words, it seems if you see some sixes and some fives in a sudoku, you can just swap them before you solve them, getting a different, but still valid sudoku. Interesting.
For the second question, thanks. I thought your parent meant something different - that the 16-clue solution was "obviously" impossible.