Recorded  FrameScore Total
--------  ---------- -----
   X          20       20

   0
   /          20       40

   X          20       60

   0
   /          20       80

   X          20      100

   0
   /          20      120

   X          20      140

   0
   /          20      160

   X          20      180

   0
   /          20      200

   X       (NOT PART OF ANY FRAME)
                     -----
                      200
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X (NOT PART OF ANY FRAME)

Not true. It's part of the 10th frame. The so-called "extra" or "bonus" ball given for a mark (strike or spare) in the 10th frame is merely a natural consequence of the scoring rules. In any frame, a strike is 10 + next 2 balls, and a spare is 10 + the next ball. All frames are scored the same, so there must be two balls thrown after a strike or one ball after a spare.

I think that the terminology is somewhat arbitrary here. But here is my reasoning.

If your game starts with X then X, the second strike is definitely part of the second frame, not the first. It affects the scoring of the first frame, but it is not part of it.

Similarly if you get a strike in the 10'th frame then you need to throw 2 more balls to find your final score. Because you are not going to throw any more frames, you get the chance to throw them right away. But I would like to classify them for scoring purposes as being no more part of the tenth frame than the second strike was part of the first frame. Certainly the scoring would not be any different if the custom was that the extra balls were thrown as part of an 11'th and 12'th (if needed) frames which were not themselves scored.

So in terms of when they are thrown, sure. They are thrown as part of the 10'th frame. But in terms of having a simple consistent mental model of what the scoring means, I find that it makes more sense to think of them as being outside the first 10 frames.

X 0 / X 0 / X 0 / X 0 / X 0 / X -> 200, should be 210
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No, it's 200. tilly's slightly unusual exegesis of the 10th frame above may have thrown you off. As you yourself say, the tenth frame is scored exactly like the previous 9 - the spare gives 10 + the next ball (X) -> 10 + 10 = 20, just as in the other 9 frames. Each frame in the game scores 20, and 20 * 10 = 200. Perhaps seeing it in a more bowling-alley-like style will help:

   1   2   3   4   5    6    7    8    9    10
   X   0/  X   0/  X    0/   X    0/   X    0/X
   20  40  60  80  100  120  140  160  180  200
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      +-----------------------------------------------+
Frame | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 |11 |12 |
      +-----------------------------------------------+
Tally |  X| X | X | X | X | X | X | X | X | X | X | X |
      +-----------------------------------------------+
Score | 30| 60| 90|120|150|180|210|240|270|300|310|320|
      +-----------------------------------------------+
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