Specializing in Three Arts?

Not what I meant. What I meant was that a combination of Ability and Puissant in a F/T, plus a Magical Focus, is a highly newbie-friendly standard pattern; it provides high power and a specialist theme without going for synergies that require a bit of arcane understanding of the rules. It also tastes a bit dull and vanilla if you do it too often. Thus, "for dummies" refers to basic newbie patterns that work just fine, but that you might want to look into moving away from as you understand the system better.

I use foci when I need to, or as you noted, "to try and solve all of [my] problems using [my] focus," particularly when I do Swiss Army Foci like the Titanos combination of Focus (Spirits) + Theurgy, or a Focus in Hermetic Geometry.

How many dice do you have to roll before double botching becomes more likely than single botching?

At least one. :smiley:

There is a significant cost in a botch when studying from vis. Namely, no experience points. Secondly, you lose all of the vis. Then there are the effects of the botch. It's completely up in the air as to what happens there, and it's doubly so for a double botch. Determining the effects of a botch is a highly subjective process, anyway. When working with Art scores of 20 or more, you're talking 4+ pawns of vis, for something that probably could be nearly replicated for two pawns of vis. If one has enough vis to go about a regular process of using it for study, they certainly have enough to go on a comprehensive spending spree acquiring all of the Tractatus on their Art that they can. I have two characters in different sagas that have a chance of getting to score of 30 (one is at 15 the other is at 22). They are both planning a purchase spree, both for their covenant libraries and personal libraries. But I'm I'm making the plans now so that the tractatus will be available when I'm able to read them. Both of these characters are extremely busy and probably can only take the time to spend 1 season per year on advancing their favored Art. One is adventurous and the other just recently took an apprentice (it was forced on him). If you have a cooperative Order in your saga with a lot of book trades, and the line seems to lean in that direction, studying from vis makes almost no sense, except at low Art scores, and even then it's questionable...

Canonically, don't nine of the Arts have Roots that can be had for a single Pawn, such that any covenant that wants a copy can have one? Primers are not hard to find and are likely to be better than personal instruction, so yeah, no reason to study low-level arts from vis unless for some reason you're off the Redcap network.

Not as sure about tractati though. Those depend much more on the Order in your saga, I think; for a Hermetic tractatus to be considered sound, it has to be made by an excellent writer, and at a high Art level you really should have better things to do with your time than study vain ones.

There are nine Roots, yes. Covenants says that they can be had for a small consideration paid to a Redcap. Later the price of summa is tied to their level, for sound books. I consider Roots to be sound books due to their quality, but the price based on level seems exorbitant. My feeling is that these books are priced at the same level as a sound tractatus, because like tractatus they might be useful for only one season L6Q21 books probably cost a bit more than L5Q15 books.
Then there are also the Branches of the Arts, and other summae. A L20Q10 summa is not out of the realm of possibility either to get on up to L20 in reasonable time. It does require the author to have had a score of 40. IF one is possession of a L20Q10 book, then one can delay study of tractatus until level 20.

A sound tractatus, as defined by Covenants is around Q11, I felt that this is probably better reflected in a range of Q9-Q11. Sound tractatus cost 2 pawns of vis (Covenants, 95). What does it take to get to Q10? I'll presume that these books are not enhanced with resonant materials, which makes trading easier, but they are all bound with skilled craftsman, so all these books start with a Quality of 6 (+3 constant + 3*1 for each of the skilled professionals), which means the author has to have Com+Virtues=4. I believe the only virtue that applies is Good Teacher, which adds +3, so a character only needs have a Com of 1. Or the person has Com of 4, which is also not a problem in a Order that has stat boosting rituals accessible. Alternatively, if you add the purchased resonant materials into the mix, you reduce the total necessary from Com+Virtues to 3. If you total the authors that have existed in the Order for a given Art, it's not unreasonable to expect that There are 26 or so sound tractatus available on an Art in existence. I mean the guy who has an Art score of 30 can write 6, 31 lets him write 7. The author with a high score and decent Com is going to write all of the tractatus he can.
Note an excellent writer probably has Good Teacher and Com 3 for a total quality of 12, or 13 if using purchased resonant materials. The best writer possible Com 5 and Good Teacher produces Q14 or 15 (resonant materials). I'm not even discussing any of these that might exist
As to the willingness to purchase vain tractatus, which would still cost 2 pawns of vis, that's probably still a better deal, if we presume a Q6 vain tractatus costing 2 pawns of vis, at art scores above 20, you're actually getting a slightly more efficient use of vis, although the time cost does go up. If you have more vis than time, it makes sense. If you have more time than vis, it doesn't. A master of an Art grinding towards a score of 40 or more probably doesn't care about the vis, but he does care about the chance of a mistake of studying from vis, which would eat up time, destroying any seasonal advancement.

Joking aside, you're avoiding my point. Double botching on two or more dice is going to be much less common than single botching on the same number of dice. At some point the math will cough up a bigger chance of multiple botches, but I assume that it's a pretty large number of dice. If the point is to avoid losing seasons of study while studying vis, reducing the botch dice to the minimum is the most effective way. A 1% chance of a season losing botch is pretty small.

Yes, studying tractatus bought with vis is cheaper and eliminates seasonal loss entirely. If you're patient and thrifty, you can eventually raise your arts at a minimum cost with almost zero risk (trading has its own risks). But there are certainly impatient, isolated and paranoid magi out there.

I wonder how many vain L20 Summae there are out there. They're still valuable at Q6-8 and they provide a slow but reliable path upwards, especially for magi with Book Learner and/or affinities in those arts. Or would those authors be better off churning out a L15, higher quality summae?

Not really. Discussing odds is fine, but the worst case scenario is a single botch, because you get nothing, without a chance of getting more. Options that provide a small chance of turning disaster into less of a disaster are important. It's true that the Twilight experience from a double botch can be bad, but you are not as likely to have a bad one at low warping scores. If my warping score is low, I want to experience Twlight, the single botch is the worst case of having a botch.
At higher warping scores, someone who has an Art score of 30+ probably has a decent warping score, one wants to void Twilight and botching at all costs, so is the character willing to accept the loss of all seasonal advancement at the cost of a loss of season and a loss of the vis? Or would he rather advance with a vain tractatus, which costs half or less of the vis and doesn't involve doing something over again.
It is small, when viewed through the lens of a single die roll, when you multiply that across the domain of die rolls a character or a player makes, it's almost a certainly to happen. It's not if one is going to botch, it's going to be when is someone going to botch and how bad is it going to be. A magus character losing an entire season is a huge, big deal.

One doesn't even have to be patient, but have some vis available and the ability to plan ahead. My magi plan ahead, to varying degrees. The impatient magi can increase the amount he spends to acquire tractatus sooner. But it's not really about being impatient and thrifty, it's more about being thrifty and risk averse. Studying from tractatus is cheaper, but it's not unreasonable believe that you could get a steady flow of books to match your study schedule, especially with a tiny bit of advance planning.
While you say that trading has its own risks, IME, SGs tend to stay out of the way of this, or perhaps introduce a story that needs to resolve something for the effort to continue...

Impatient magus: I need 3 tracatus now
Redcap: I can get them to you next season for a cost of 10 pawns
Impatient magus: Well, I only need one next season, so how about I pay you 3 for that tractatus, and bring the others to me over the next seasons at the standard rate, and I'll add another pawn for you to keep your eyes out for other titles on (Art) for me and let me know when they become available.

Smart magus who plans ahead: Redcap, let's work up a contract, where you deliver 1 sound tractatus per year to me over the course of the next 10 years, and I'll give you 25 pawns of vis now to cover the cost and any personal expenses. Failure to deliver a tractatus in a year will trigger a refund of 3 pawns of vis to me.

Adjust numbers as seem reasonable to your saga.

Actually rolling, so after any reductions, roughly 12.75 to get there, so 13 dice. I don't have a solver on hand and the free online ones are failing me. I had to estimate graphically. The inequalities are 1>(9+2x)*9^(x-1)/10^x and x>0, x being the number of dice rolled. Note that this is for double+ botching (so double, triple, quadruple, etc.), which is what I figured the question really was about.

When rolling d botch dice the probability of getting:

  1. exactly one 0 (single botch) is d*(0.9)^(d-1)*0.1
  2. exactly two 0s (double botch) is (d*(d-1)/2)(0.9)^(d-2)(0.1)^2
    The ratio between the first and the second probability is then 0.9/(0.1*(d-1)/2)=18/(d-1).
    Thus, a double botch is exactly as likely as a single botch when rolling 19 botch dice; less likely with fewer dice and more likely with more.

Of course, you need fewer dice to make a double-or-worse botch more likely than a single botch! As callen wrote, a double-or-worse botch is more likely for 13 or more dice, less likely for 12 or fewer.

Back to the OP:

The catch here is that you have a focus, and it seems you only want to specialize within the focus (as opposed to a character specializing in e.g. Creo Ignem, who's certainly awesome at Creo Ignem, but also good at non-Ignem Creo and non-Creo Ignem). Because the focus doubles your lowest Art, your best strategy to maximize your TeFo(Fo) total for a given amount of experience is to keep all three Arts as evenly matched as possible -- this would be true even if you had the Affinity and Puissant Virtues applying to the Technique but to neither of the Forms, unlike what was claimed elsewhere in this thread. One consequence is that Affinity & Puissant aren't as big a help, because they each apply only 1/3rd of the time, so to speak -- as was pointed out, some more "general" xp boosting Virtues are probably a more effective investment.

No, this is not true. This holds if you lack the Affinity or Puissant but fails when you include them. Let's say you have Affinity w/ Technique and with neither Form (F1, F2 below). Here is a counter-example to your claim about maximization holding with Affinity:

Te 12, F1 12, F2 12 -> 12+2x12=36 at a cost of 52+2x78=208
Te 14, F1 11, F2 11 -> 14+2x11=36 at a cost of 70+2x66=202

The latter gets you the same TeFo(Fo) for 6 fewer experience points. Those 6 points are more than half of the 10 points needed to reach Te 15. And this is using Affinity at its weakest. Really, we should treat Affinity closer to +55% to +60% or so, the specifics varying by style and age.

Edit: Now that I have more time, here is a similar counter-example with Puissant Technique:

Te 9+3, F1 12, F2 12 -> 12+2x12=36 at a cost of 45+2x78=201
Te 11+3, F1 11, F2 11 -> 14+2x11=36 at a cost of 3x66=198

Again, keeping them balanced does not result in as high a TeFo(Fo) combination for the same experience as having them slightly off-balanced in favor of the singleton with Puissant.

Now, if you want to argue that Affinity/Puissant lose a lot of their value paired with Focus in such as case as many of us have said, that is showcased by the relative balance that is still desired.

If we look at that same example with Affinity:

Te 12, F1 12, F2 12 -> 12+2x12=36 at a cost of 52+2x78=208
Te 14, F1 11, F2 11 -> 14+2x11=36 at a cost of 70+2x66=202

and compare it to the best without Affinity

Te 12, F1 12, F2 12 -> 12+2x12=36 at a cost of 3x78=234

So the Affinity has saved 32 points (again, going with worst-case 50% bonus) out of the 234 that would have been spent or roughly a 13.6% reduction. Meanwhile, Affinity with just a TeFo pair with the standard for-dummies approach has roughly a 16.7% reduction in the equivalent case. Plus, Affinity is already generally worse than the general learning Virtues over enough time, so a further reduction isn't helping it. That's why several of us were saying spreading things out makes things like Book Learner even better choices. Even things like Skilled Parens don't lose their value and so improve relative to Affinity similarly.

Note that the optimal point is when both Arts cost the same to advance:

  • advancing Form at score 11 costs 8 xp with Affinity,
  • advancing Technique at score 7 costs 8 xp without.
    In this case, you'd want Te = 8 for Fo = 12.

The same applies for 3 Arts:

  • advancing both Forms at score 8 costs 6 xp each with Affinity, for a total of 12 xp
  • advancing Technique at score 11 costs 12 xp without.
    In this case, you'd want Te = 12 for Fo1 = 9, Fo2 = 9.

Puissant does not change the calculation:

  • Te = 8 for Fo = 12+3.
  • Te = 12 for Fo1 = 9+3, Fo2 = 9+3.

Now with a focus they have to be equal or you lose 2 points. As you see in the example above, 12 = 9+3 so the balance is perfect at that point. If you don't have Puissant:

  • 44 xp gives you 12+12*2 = 36
  • 55 xp gives you 14+11*2 = 36
    The optimum without MMF would only require 41 xp for 14+10 = 24, so the waste is only 3 xp.
    {note that this is different from callen, since I used 2 Affinities on the Forms whereas he used one on the Technique.}

tl;dr
Always advance the Art that requires the fewest xp first, but don't forget that advancing both Forms costs twice as much.
If you have MMF, give Affinity to the twin Forms and keep all 3 Art scores equal.

While most of what you say is good, your final bit of advice is wierd:

That is not optimal for two points of Virtues. For example, let's look at the combined score of 36 again. With your suggestion:

Te 12, F1 12, F2 12 -> 36 for a cost of 78+52+52=182

Now try it with Affinity + Puissant in the Technique:

Te 13+3, F1 10, F2 10 -> 36 for a cost of 61+55+55=171

Attaching two points of Virtues to the singleton is more effective than attaching one to each of the pair. Sure, if you compare Affinity in the singleton to Affinities in each of the pair, the latter is better. But the same number of Virtues have not been spent and this is why the latter is better.

Additionally, there is another point there to be adjusted. We remember we are restricted to two Affinities. You might argue this is the point of putting them in the pair. But why an Affinity in each of the pair? Why not one in the singleton and one in the pair? If you keep everything balanced, this clearly saves just as much as putting them in the pair. But now you don't have to keep things balanced, being able to go for a slight imbalance as I've been showing is more optimal. Here is something more optimal with two Affinities, this time putting one in the singleton and one in one of the pair:

Te 14, F1 11, F2 11 -> 36 for a cost of 70+44+66=180

Note that this has saved two experience over your suggestion of putting them in each of the pair.

Here are the best options between Affinity and Puissant (assuming high enough levels that Affinity > Puissant and assuming weakened Puissant of only +50%) for TeFo(Fo) or Te(Te)Fo with Focus:

1 Virtue: Affinity in the singleton
2 Virtues: Affinity + Puissant in the singleton (what several of us suggested above)
3 Virtues: spread out two Affinities and one Puissant, one to each Art
4 Virtues: Affinity + Puissant in the singleton, Affinity in one of the pair, Puissant in the other of the pair

But remember that these Affinities and Puissants are not as strong as generally due to the distribution of experience, while general experience bonuses can be more effective. So you may well be better off putting points into things like Book Learner. Also, advancing three Arts is slower, so it takes longer for Affinity or Puissant to become better than something like Skilled Parens than it might otherwise.

Alas, you are right. Keeping everything balanced would be the best strategy if one had no Affinity/Puissant in the Technique (regardless of whether one had Affinity/Puissant in either or both or neither of the Forms). But with Tech+2Forms AND an Affinity/Puissant in the Technique it pays (a little) to raise the Technique (a little) higher.

My mistake was assuming that:
"If one Art is lower than the other two (with no accumulated xp) the best strategy is to raise that Art", which is true regardless of Affinity/Puissant, implied:
"The best strategy is to keep everything evenly matched", which is in general false.

I determined the optimal strategy for TeFo(Fo) with Focus and an Affinity, though I think I have a small typo in my experience calculation code:

Fo=(4*Te-1)/6

However, this did not include the restriction that each have to be integers, which makes things wonky. For instance, with 202 points, you want Te 15.7846 and Fo 10.3564 for a combined total of 36.4974. But the reality would be that those are 15 and 10, not quite having reached 16 and 11. So the actual combined total would be 35 instead of 36. It is better to go with Te 14 and Fo 11 for a combined total of 36. But here's the tricky part. What about with 190 points? You can get to Te 15 and Fo 10 for a combined total of 35 already. So what do you do when you're sitting at Te 14 and Fo 10? If you raise the Technique, you'll move up from 34 (to 35) faster. If you raise both Forms you sit at 34 longer but get to 36 faster. Thus the "wonky" part. The ideal choice would then more likely be based on what else you want to do with them. For instance, Fo 11 gives you an extra +1 Magical Defenses in two areas, so I would go for the 11's first unless I really had a pressing other TeFo combination begging me to raise the Technique.

Anyway, there is the formula you want if you have an Affinity in the singleton only.

Also note, you get the same optimization formula if you include both Affinity and Puissant with the singleton. However, to be specific, that Te value would be before the +3. So ignore Puissant to figure out the optimal values, and then add Puissant back in to see what combined total you get.

Yes, I missed that. OTOH 2 xp aren't worth the complexity, but I wonder if this holds true at higher scores.

Te 17, F1* 17, F2* 17 -> 51 for 153+102+102 = 357 xp
Te* 18+3, F1 15, F2 15 -> 51 for 114+120+120 = 354 xp
Te* 21, F1* 15, F2 15 -> 51 for 154+80+120 = 354 xp
Te* 19, F1* 16, F2 16 -> 51 for 127+91+136 = 354 xp
{* with Affinity}

Neat, I hit straight on the reverse point where 2 Affinities become better than Affinity+Puissant.

Let me amend my tl;dr:
tl;dr
Always advance the Art that requires the fewest xp first, but don't forget that advancing both Forms costs twice as much.
If you have MMF, always advance the Art that requires the fewest xp first. (advancing both Forms equals advancing Technique by 2)

Yes, Affinity in the singleton and Affinity in one of the pair always beats out Affinity in each of the pairs if we're looking at the combination we're looking at. That is necessarily true because keeping them all balanced makes all the Affinities equal, but having one in the Technique allows for an imbalance in your favor. I could work out the optimal ratio later. But it doesn't seem worth the effort since there isn't a very big savings.

Yes, that will always happen eventually. But it usually takes a very large amount of time to get there after character creation. Otherwise we'd all drop 2x Puissant Art in favor of Book Learner and Independent Study every time. Similarly, no one would ever take an experience Virtue because Affinity, Puissant, and the learning Virtues all beat them out in the long run. But, again, there is the issue of being ahead for the first 10-20 years helping you get to the later years and saving you some seasons on the early end.

Oh that's easy:

  • 96 -> 98 costs {(97+98) * 2/3 = 130 xp}
  • x-1 -> x costs {x + x*2/3 = 5/3 x = 130 xp -> x = 78}
  • 77 -> 78 costs {78 + 52 = 130 xp}

y = (2x+3) * 2/3 * 3/5 = (2x+3) * 2/5 ~ 80%