Permitting six or seven challenges to acquire the title of Archimagus does not mesh well with the stated number of archimagi in Ordo Hermei (around 80 out of 1200). To demonstrate this, let us consider an example, and suppose that at some point, there was only a single archimagus in Ordo Hermei. This is not a good supposition, but it is the most conservative approach, and thus will demonstrate the issues.

Suppose that this single archimagus bestowed the title on six other magi who successfully challenged him, bringing the total number of archimagi in Ordo Hermei to seven:
1 + 6 -> 7

Now, let us next suppose that each of the six "second-generation" archimagi each were more conservative and/or challenging, and each only bestowed the title on five successful challengers. This adds another thirty archimagi, bringing the total to thirty-seven:
1 + 6 -> 7 + 30 -> 37

Let us also suppose that the third generation were more conservative or challenging still, and each of the thirty latest archimagi only bestowed the title on four other magi. This would add 120 archimagi, bringing the total to 157:
1 + 6 -> 7 + 30 -> 37 + 120 -> 157

We already have about twice as many archimagi as are suggested ("fewer than 80"). Granted, by this time some of the original archimagi may have demised, but even if the seven oldest members demised, that would still leave 150 archimagi, and even if one assumes that the thirty "third generation" archimagi demised soon after granting their final title, that would still leave 120 archimagi, far more than is suggested by GotF.

Note also that this assumes a conservative reduction in how many titles are actually granted; leaving it at six per archimagus would result in a far greater expansion (1+7+42+252 = 302 archimagi) in only four generations.

Clearly, the limit of six titular grants per archimagus is excessive, even if one assumes that archimagi don't lead particularly long lives. One does need to consider that not all archimagi would grant six titles, but even if some do, the issue arises. One also must consider it unlikely that the title of archimagus began with only a single magus; more probably, a small group of powerful magi began the tradition. Depending on how many one supposes were in that small initial group, the problem of expansion becomes much greater: even two initial archimagi double the resulting numbers, regardless of which other assumptions one uses. Three or four initial members, which is not an unreasonable supposition, results in even greater expansion.

Thus, unless one assumes that some significant number of archmagi are never successfully challenged, allowing six or seven challenges per archimagus is excessive. One could assume that such was the case initially, but that as the tradition became more formal, the number of allowable challenges was reduced, but even so, that is consistent with the example given above, and does not resolve the resulting issues. One resulting issue with a large expansion is that losing more than six challenges becomes very improbable, and thus there is little basis for a stigma to arise.

To resolve the issue, one could either change the number of allowable challenges, or vastly increase the number of archimagi in Ordo Hermei. The latter approach is undesirable for many reasons, not the least of which is the lack of motivation existing archimagi might have for allowing such a large number of magi to bear the title. If the distinction is to remain a distinction, it must retain a degree of exclusivity. Thus, existing archimagi have a motive for restricting the expansion of archimagi, rather than allowing a vast expansion.

Considering a counter-example may be useful. Let us suppose that three powerful magi began the tradition, and only bestowed the title on two challengers each. If one assumes that the tradition began fairly soon after the founding of Ordo Hermei, say around 800 AD, and if one assumes that each "generation" of archimagi lasts about one century, one gets the following results:

This rough doubling of the number of archimagi every century fits the assumptions given above, and results in around eighty archimagi by 1220 AD (if one also assumes that previous generations demise fairly soon after granting the second title). Depending on other assumptions, such as the number of archimagi that demised during the Schism War, one could allow for a greater number of challenges at some times and still obtain the results shown above.

However, the assumption that previous generations of archimagi demise soon after granting a second title is itself not reasonable. Some may demise, certainly, and some archimagi may demise before ever granting a title. Nonetheless, some archimagi are likely to live long enough to increase the total number of archimagi beyond that of only the latest generation. Thus, assumptions around the probable lifespan of magi are relevant, and can result in widely varying results.

Further, the previously noted motive for archimagi to maintain the distinction of the title by restricting the total number of archimagi remains. Thus, one can assume that many archimagi will seek to make their challenge as difficult as possible, perhaps even seemingly impossible, in order to defer granting any title.

Some countervailing factors arise, however. For example, it is canonical that some archimagi have been "invited" to challenge for the title. In other words, occasionally at least some archimagi want to grant the title to a specific maga (such as was the case with Prima Bilera Guernici). A second issue arises when one considers that it is traditional to challenge one's own parens first, if one's parens is an archimagus. One would expect that many archimagi would desire to grant the title to a filiius more than to some other random maga, and thus challenges might be arranged so that one's filii would possess an advantage in any challenge.

All of these factors could vary, and thus any model of generational expansion of archimagi could have widely varying results, depending on the weight one assigns to each factor. In the end, the stated number of eighty archimagi at 1220 AD is the goal, and restricting the permissible number of challenges makes that result more likely, regardless of how other factors are weighted. The Saga Rule allows for three or four challenges per archimagus, more than sufficient to justify eighty archimagi in 1220 AD, regardless of other factors.