F.L.A.M.E.S. -- is it fair?

This post is "inspired" by the so called FLAMES algorithm.

For the uninitiated, this is a very simple exercise. Take names of two people, usually of opposite genders and cross out all the common alphabets between the two names. Post that, count the number of alphabets that remain and get that number. Now write F L A M E S on a piece of paper and start counting from F to S, then go back to F and again the same process is done, with the number you have. Every time the count ends, cross out that alphabet, and restart counting from the next alphabet. Repeat this till you cross out five of the alphabets and remain with one....

I've never encountered this before even though it seems to be common knowledge, maybe this was for the better. Anyway, this is a curious little calculation game and out of curiosity as to whether this "game" is really fair or whether it has something about it that makes it unpredictable, I did a little empirical investigation.

Terminology
Mismatched letters: If the names are GEORGE W. BUSH and NELSON MANDELA. Then the number of mismatches is the number of letters that do not appear in both names. Here GEORGE W. BUSH and NELSON MANDELA, so the number of mismatches = 7 + 10 = 17. For simplicity, lets call the number of mismatches M. With that background, I generated the outcomes and the sequences by which they arise for every M in the range 1 to a 1000.

Sequence: The sequence in which the letters of FLAMES are struck out, i.e. M then A then L then F then E then S, etc.

Some interesting findings(yes, yes, I dont have a rigorous proof (as yet) but the following seem to hold true for M = [1,1000])

• Observation - 1: There are only 60 unique sequences for any value of M. Why is this interesting? Since there are 6 letters in FLAMES, the number of possible sequences by which each of these letters can be struck out is 6! = 720. But of these only 60 can occur.
  
  
• Observation - 2: All of these unique sequences occur in the range M = 1 to 60. This means that every M in the range 1 - 60 has a unique sequence associated with it. From M = 61 onwards, the sequences begin to repeat. This is probably why the game seems so unpredictable as most reasonable pairs of names would have M in this range and at every step it seems like you cannot predict what outcome will emerge.
  
  
• Observation - 3: For M greater than 60, the sequences begin to cycle with period 60. The sequences for M = 1, 61, 121, .... are identical and so are the sequences for M= 2, 62, 122, 182,.., for M = 3,etc. In fact this is true for all M in the range 1 to 60 where the sequences for any M' = 60*n + M are identical (where n belongs to the Natural numbers).
  [I dont have a proof of this beyond exhaustive checking]
  
  Observation - 4: Each of the six possible outcome is equally likely in the range 1-60. (see Figure 2 below). So, if pairs of names can uniformly lie in the M = 1 to 60 range, then the outcome is "fair". But if we look at specific ranges then it is not so fair (see Figures 3 onward).
  

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Figure 1: Variation in predicted outcome for different numbers of mismatched letters. Looks arbitrary but is it so?

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Figure 2: Percentage of each possible outcome for numbers of mismatched letters over the entire range 1 to 60. Surprise -- every outcome is equally likely in this range!!!

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Figure 3: Percentage of each possible outcome for numbers of mismatched letters varying from 1 to 10. Definitely skewed towards 'E'

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Figure 4: Percentage of each possible outcome for numbers of mismatched letters varying from 11 to 20. Absolutely no 'S'

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Figure 5: Percentage of each possible outcome for numbers of mismatched letters varying from 21 to 30. Absolutely no 'L'

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Figure 6: Percentage of each possible outcome for numbers of mismatched letters varying from 31 to 40. Absolutely no 'E'

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[More updates on this investigation shortly...]
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