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.
"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?