User:Stupac2/math

Ascension Math

The goal of this section is to teach those interested a little bit about the kind of math people do to figure out what's optimal in an ascension. Oftentimes I get asked questions that could be answered with a bit of research and math by the person asking. I don't particularly mind this, but I do believe it's better for people to answer these questions themselves. To that end I'm going to run through a few illustrative examples that will hopefully gives the reader an idea of how to answer the "Which is better..." question for their own situation (because it's always situation-dependent). If you're particularly allergic to math, this section will probably not interest you.

Also, out of laziness, I'm not going to link to the wiki pages too much. If you don't know what something is, wiki searches should find it easily.

Powerleveling vs Gongs

Inspired by a comment by Pantsless, this section will calculate the stats/turn of powerleveling in spookyraven (as a moxie class) to using a gong to get roachform (and unpopular).

The place to begin is to figure out what each one is giving you, step by step. Form of the Roach takes 3 turns, 2 of which gives you ~mainstat (hereafter simply M) in mainstat, and the third of which should be unpopular (+30 ML for 20 turns). So you get 2M + 30*20/8 mainstat, assuming 20 successful combats in a row after the gong use (not normally possible, but we're examining best-case here and you can do that in some places, such as the battlefield or the cyrpt).

But you also need to factor in using a llama over using a bandersnatch. Getting a gong takes 5 turns (on average, when doing this math you always use averages). We need to make some assumptions about familiar weight, to make it easy I'm going to assume both are 25 lbs. The bander would give 25 mainstat in that time, and the llama (as a 1/2-weight volley) would give .5*sqrt(weight/2) stats, which at 25 lbs comes out to 1.76 mainstat/turn (for this analysis we don't care about offstats). This comes out to ~8.8 stats total. So using the llama loses you 16.2 mainstat.

So the gong gives (2M+75-16.2)/3 stats/turn. Now we need to figure out spookyraven.

This is where it gets fun. The ballroom has a native combat rate of 80%. The only normal noncombat is curtains, which gives ~M moxie for 1 turn. So if you have N noncombat modifiers running, you get (.2+N)*M mainstat/turn from curtains, on average (N should be .2 in most softcore ascensions). Monster level in the ballroom is 110, which means you get 13.75 mainstat from each combat, or (.8-N)*13.75 on average.

Right now it looks like gongs win easily. But this is missing a few things: ML and stat modifiers on you, and dance cards.

The former are easiest, if you're running S stat modifiers they give S/2 mainstat, and if you're running ML ML, you get ML/8 mainstat. We simply add those into the combat stats for (.8-N)*(13.75+ML/8+S/2) stats from combat/turn.

Now dance cards. They have a 15% drop rate, meaning you'll get .15*(1+I) per waltzers combat (I is +item). Normally you encounter waltzers on 1/3 combats, for (.8-N)*.15*(1+I)/3 dance cards/turn. With olfaction they have a 77.5% encounter rate, meaning you get (.8-N)*.15*(1+I)*.775 dance cards/turn. Dance cards give ~2M moxie, meaning dance cards give you 2M*(.8-N)*.15*(1+I)*.775 stats/turn.

Putting this all together you get (.2+N)*M + 2M*(.8-N)*.15*(1+I)*(.775|.333) + (.8-N)*(13.75+ML/8+S/2) stats/turn from the ballroom.

You'll notice that comparing this mess to the gongs depends really, really heavily on what your exact situation is. This is why I want to encourage people to think this way so they can figure these questions out for themselves, because it's damn near impossible to give a good answer without knowing everything about a person's situation.

Anyway, let's fill some of these things in. S should be 6, from putty hat and pilgrim shield. ML will vary a lot, but let's assume level 8 (so 16 from Ur-kel's), and 10 from radio. We'll also assume native 5-lb fairy, leash, empathy, sympathy, pet-buffing, and equipment for 30-lbs. And then 35% passive items, phat loot, knob goblin eyedrops, 2 Jr.s and the pink shirt. We'll also assume -20% combats (N = .2). Finally, we'll assume olfaction. So let's put this all together.

.4*M+2M*(.6)*.15*(1+1.917)*.775+.6*(13.75+3+3.25) = .4M+2M*.203+12 = .8M + 12.

Now we can compare this to roachform, which gives .667M + 19.6. These will be equal at ~57 mainstat. Which means from 58 mainstat on (about 1/3 of the way through level 8) the ballroom beats llama gongs, under all of these assumptions (which will not always be true).

Bandersnatch vs. Sombrero

This question comes up from time to time, and fortunately I've already answered it. But let's walk through the math.

So we already know that a tuned bandersnatch gives sqrt(weight) mainstat. A sombrero gives sqrt(sombrero weight)*(1 + sqrt(monster level - 4)) / 10. stats, half of which will be mainstat. So right now let's ask our question: At what ML will a sombrero beat a bandersnatch assuming equal weight?

The first step is to set the two formulas equal, then we get: sqrt(W) = sqrt(W)*(stuff). So let's cancel out those sqrt(W)'s and just have (stuff) = 1.

So we now have: 1 = (1+sqrt(ML-4))/20. This leads to: 19 = sqrt(ML-4), which then leads to ML = 365. So if you have the two familiars at the same weight, the sombrero beats the bandersnatch at ML = 365 (which you can encounter in some bosses and late in the game with massive +ML). But these assumptions are bad because the bander will be much heavier after a whole ascension. This is why most speed ascenders have nearly completely eschewed the sombrero.

Others

Those are the two things that occurred to me to demonstrate the basic idea behind ascension math. If you can think of some others that are illustrative (and not things you want me to calculate out for you) let me know and I'll consider adding them.