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.
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.