I think they addressed this in the edit they made:

>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.

The implication (not entirely clearly stated in the article) is that if a puzzle with n clues is solvable, a puzzle with n+1 clues is also solvable.

Conversely, if a puzzle with n clues is unsolvable, any puzzle with n-1 clues is similarly unsolvable.

I don't get where the article makes this assumption:

"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?

For the first question, about why 123 456 7 doesn't have a unique solution:

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.

For the first question: thank you, I didn't realize that the article meant that those two numbers are supposed to be swapped THROUGHOUT! (every occurrence in the grid, like find-and-replace). This makes sense to me. Likewise, it seems to me true that you can swap-and-replace any two numbers in any completed graph. (Really, they're just symbols, it could be turning the original numbers into A B C D E F G H I in the first step - then you can map these 1:1 to one through nine in the second step however you want) and, therefore, you could do that operation on the original clues and get a valid sudoku as well.

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.

Regarding 7 hints provided. Suppose you have a puzzle which contains 7 digits for hints. Assumming the provided digits are 1-7 placed around the grid and that they permit you to produce a unique solution for those digits (that is you can fill up all but 18 spaces). If you then fill in those remaining spaces with 9 8s and 9 9s, then, however you arranged them, the opposite arrangement is valid.

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.

thanks. it's like why a system of three variables for which you have two equations (like x = 2y + z and x = 3y + 2z + 7) isn't solvable.