Alright, here we go. If our strategy is to grind to MW 15 + 3 or 4 cp before calling the Amanuensis and then take the challenge at MW 15, how many actions can we expect to spend going from 0 Notability to 5?
Some initial numbers: MW 15 + 3 cp is 123 cp total, MW 7 is 28 cp, and we’re assuming we can grind at 2 cp/action. So going from MW 0 to 15 is 62 actions, and 7 to 15 is 48. Add 2 to both of these to summon and play the Amanuensis for 64 and 50 actions to try again after success or failure, respectively. Finally, I’m assuming BDR = 20 so the challenge difficulty is exactly 6*N.
Going from 0 to 2 Notability at BDR 20 is entirely straightforward, so worst case you’ll spend 128 actions if you somehow need to summon the Amanuensis for both of these but it will probably be a lot less. Ultimately it depends on your deck and calculating an expected actions to Notability 2 just isn’t worth it, so we’re going to assume you can get there without much trouble.
The next step is going from 2 to 5, which is harder. At MW 15 2 to 3 is a 75% challenge, 3 to 4 is a 50% challenge, and 4 to 5 is a 37% challenge. So let N(x,y) be the expected number of attempts to go from Notability x to Notability y. Then…
N(2,5) = 1 + .75N(3,5) + .25N(2,5) #We always spend an action, but only advance 75% of the time
N(3,5) = 1 + .50N(4,5) + .50N(3,5) #Same idea, different numbers
N(4,5) = 1 + .37N(5,5) + .63N(4,5) #And again
N(5,5) = 0 #It of course takes 0 actions to go from notability 5 to notability 5
Working backwards to solve all this, we get N(2,5) = 8.7 Or at least I get that, someone should double check. So it’s going to take let’s just say 9 attempts to go from Notability 2 to Notability 5, but how many actions is this?
Recall that once we got to Notability 2 we had to spend 64 actions to get MW back to 15. Similarly, once we hit Notability 3 it will take another 64 actions to build it back up, and the same for when we hit Notability 4, so three of our attempts cost 64 actions. Our other attempts were all failures, though, and failures only require us to spend 50 actions to get back to MW 15 since we start at 7.
So using this strategy we can expect to make 3 attempts costing 64 actions and 6 attempts costing 50, for a total cost of 492 actions to go from Notability 2 to 5, plus whatever it takes to get to Notability 2 in the first place. So absolutely worst possible case is 620 actions, and around 530 might be a more reasonable guess.
Obviously, getting lucky on your first 37% challenge or bombing a 50% challenge twice in a row will swing this pretty far from that number in either direction, but as far as my math skills go (and feel free to tell me I’m doing something hilariously wrong, it’s very possible) that’s what you should expect when you begin the grind.
tl;dr - This strategy should take you from no notability to notability 5 in about 530 actions.