[pdurrant]

Originally Posted by GeoffC

Spoiler:

Night 1

X+Y = 13 or 10.

East sees X, therefore he knows West must be seeing (13-X) or (10-X). If X ≠ 13 then no one is freed and X ≠ 0.
West sees Y, therefore he knows that East must be seeing (13-Y) or (10-Y). If Y ≠ 13 then no one is freed and Y ≠ 0.
No one is released, therefore X ≠ 13, Y ≠ 13, nor 0.

Night 2

Neither X nor Y = 0 1 2 11 12 13
East sees X, therefore he knows West must be seeing (13-X) or (10-X).
West sees Y, therefore he knows East must be seeing (13-Y) or (10-Y).
If neither X nor Y = 10 neither can be certain.
No one is released, therefore X ≠ 10, Y ≠ 10, nor 3.

Night 3.

East sees X, knowing West sees (13-X) or (10-X).
West sees Y, knowing East sees (13-Y) or (10-Y).
If neither X nor Y = 11 neither can be certain of what the other sees.
No one is released, therefore X ≠ 9, Y ≠ 9, nor 4.

Night 4.

East sees X, knowing West sees (13-X) or (10-X).
West sees Y, knowing East sees (13-Y) or (10-Y).
If neither X nor Y = 10 neither can be certain of what the other sees.
No one is released, therefore X ≠ 8, Y ≠ 8, nor 5.

Night 5.
Only 6 & 7 remain = 13

Sorry, you have the right answer, but your reasoning seems to be incorrect.

Spoiler:

After Night 1, and if the wise men still prisoners, you seem to be saying that it is not possible for X or Y to be 0. It is. Consider X=10 and Y=0. Neither can give a certain answer on the first night, but Y=0.

But I think you're beginning to see the way to the solution.