Finding the unique solution to the initial challenge is fairly straight forward (once I'd reminded myself of which end is which on a g++ compiler)
*** START POINT:
41 92 5 58 . . 3 . 45 26
. . 42 . . . . . . .
. . 99 64 . . 23 56 27 .
66 . . 89100 . 54 . . 1
39 . 67 . 87 60 . 62 . .
. 69 88 95 84 71 76 . . 19
. . 81 . . . . . . .
. 9 . 83 . . . . . 49
37 . 11 . 35 78 73 16 . 30
. 13 . . . . . 31 50 .
***
Found 1 solutions!
41 92 5 58 43 24 3 22 45 26
6 65 42 91 4 57 44 25 2 21
93 40 99 64 59 90 23 56 27 46
66 7 94 89100 63 54 47 20 1
39 98 67 70 87 60 85 62 55 28
8 69 88 95 84 71 76 53 48 19
97 38 81 68 77 86 61 72 29 52
12 9 96 83 80 75 32 51 18 49
37 82 11 14 35 78 73 16 33 30
10 13 36 79 74 15 34 31 50 17
How does AOC++ work - do you want my workings (code) too, or just the stdout?
In terms of trying to remove clues from the puzzle while still keeping a single unique solution, well....
At least the first seven steps (that is, Knight moves 3, 5, 9, 11, 13, 16 and 19) can be removed while the puzzle still has a single unique solution. Trouble is my current solution finder is much worse than exponential search time for the number of contiguous unknown moves, so I need to rethink my approach to find exactly how many can be removed. If all else fails, I'll have to dig out the idiot's guide to MapReduce.
This is by no means the complete solution, but the best I've got so far...
So far, I can blank 16 squares, so down to 28 'clues' (down from 44 in the OP) and it still only has one unique solution.
Found a puzzle with 28 clues that still has exactly one solution!..
. 92 5 58 . . . . 45 26
. . 42 . . . . . . .
. . 99 . . . 23 . . .
66 . . 89 . . 54 . . 1
39 . . . 87 . . 62 . .
. 69 . 95 84 . 76 . . 19
. . 81 . . . . . . .
. 9 . . . . . . . 49
. . . . 35 . 73 16 . 30
. 13 . . . . . . . .
==
Update1: got it down to two slightly different puzzles each with 26 filled squares that have a single solution.
Found a puzzle with 26 clues that still has exactly one solution!..
. 92 5 58 . . . . 45 .
. . . . . . . . . .
. . 99 . . . 23 . 27 .
66 . . 89100 . 54 . . 1
39 . . . 87 . . 62 . .
. 69 . 95 . . . . . 19
. . 81 . . . . . . .
. 9 . . . . . . . 49
. . . . 35 78 73 . . .
. 13 . . . . . 31 . .
AND
Found a puzzle with 26 clues that still has exactly one solution!..
. 92 5 58 . . . . 45 26
. . . . . . . . . .
. . 99 . . . 23 . . .
66 . . 89 . . 54 . . 1
39 . . . 87 . . 62 . .
. 69 . 95 . . 76 . . 19
. . 81 . . . . . . .
. 9 . . . . . . . 49
. . . . 35 . 73 16 . 30
. 13 . . . . . . . .
Update2: Down to 25 clues (so 19 blanked vs OP)
Found a puzzle with 25 clues that still has exactly one solution!..
. 92 5 58 . . . . 45 26
. . . . . . . . . .
. . 99 . . . 23 . . .
66 . . 89 . . 54 . . 1
39 . . . 87 . . 62 . .
. 69 . 95 . . 76 . . 19
. . 81 . . . . . . .
. 9 . . . . . . . 49
. . . . 35 . 73 . . 30
. 13 . . . . . . . .
[/spolier]
So my best results:
a) count(n) = 21
b) sum(n) = 1223
(a)
Found a puzzle with 23 clues that still has exactly one solution!..
. 92 . 58 . . . . 45 .
. . . . . . . . . .
. . 99 . . . 23 . . .
66 . . 89 . . 54 . . 1
39 . . . 87 . . 62 . .
. 69 . 95 . . 76 . . 19
. . 81 . . . . . . .
. 9 . . . . . . . 49
. . . . 35 . 73 . . 30
. 13 . . . . . . . .
(b)
Found a puzzle with 26 clues that still has exactly one solution! sum = 1223
. 92 5 58 . . 3 . . 26
. . . . . . . . . .
. . 99 . . . 23 . 27 .
. . . . . . 54 . . 1
39 . 67 . . . . 62 . .
. . 88 . 84 71 . . . 19
. . 81 . . . . . . .
. 9 . . . . . . . .
37 . 11 . 35 . . 16 . 30
. 13 . . . . . 31 . .