I wanna go home

ruleset/forGameMasters/balancingConsiderations/spellcasting.md

@@ -38,7 +38,9 @@ Doing something like this discourages the indiscriminate use of their big spells
 ### How many "encounters" do we expect players to have in a day (long rest)
 Now the word encounter here is very loosely defined, this could be anything that requires resources to be used. Some form of puzzle, trap, environmental hazard, or combat encounter. All the will assume players are short resting after each combat encounter which are interspersed with minor encounters like traps, puzzles, and social encounters.
 
-That said, for this we can just pull some numbers straight from the deepest reaches of our asses and say 2 to 3 'minor' combat encounters with a boss or mini-boss fight, or 5-6 combats without a boss fight, and about 2-3 non-combat encounters between each one
+That said, for this we can just pull some numbers straight from the deepest reaches of our asses and say 2 to a 'minor' combat encounter with a boss or mini-boss fight, or 3 combats without a boss fight, and about 2-3 non-combat encounters between each one
+
+>I did have this set at 6 encounters/long rest previously, but chose to go with 3 combats and group the non-combats up because if we are accounting for every roleplay encounter, 6 is too few
 
 ## Now for the math
 With all of this in mind, its time to pull some formulas from the deepest pits of hell.
@@ -63,57 +65,71 @@ each character isn't expected to cast spells higher than a certain tier based on
 #### How many spells will a player cast
 lasting on average 4 rounds, a player will have time to cast at most 12 spells per combat encounter, 4 can be cast as advanced actions, 4 as basic actions, and 4 as reactions. No spell should be made that does not cost one of these actions.
 
-12 spells per encounter is a bit extreme, but lets budget for one spell per round per combat encounter, meaning on the extreme end of 6 combats that is $6*4=24$ spells between each long rest just on combat encounters, adding two spells across the 12-18 minor encounters (remember, these could be as simple as talking to an NPC or solving a puzzle) brings us to 26 spells between each long rest (complete SP reset)
+12 spells per encounter is a bit extreme, but lets budget for one spell per round per combat encounter, meaning on the extreme end of 6 combats that is $3*4=12$ spells between each long rest just on combat encounters, adding two spells across the 12-18 minor encounters (remember, these could be as simple as talking to an NPC or solving a puzzle) brings us to 26 spells between each long rest (complete SP reset)
 
 #### How many SP will those spells take
 The exact amount will vary, but our goal should be for their 'small' spells average 3 'tiers' lower than their 'high tier' spells. Say of those 26 spells, 2 are their 'high tier' spells, this leaves a level 10 character casting 24 tier 2 and 2 tier 5 spells between rests, which means we need to do math depending on our level ranges from before
 
 Our function looks a little like this, where H is the SP cost of a high-tier spell for the level and L is the cost of a low-tier spell, and $f(x)$ is the characters level
 
-$f(x) = (2*H) + (24 * L)$
+$f(x) = (2*H) + (12 * L)$
 
 This is the simple version, where H and L are precalculated, it can probably be expanded as a function which only needs a characters level, but we have no need for that equation right now. At least for now, if I decide I need it later I suppose I will be stuck doing some more math
 
-|Level|$2*H$|$24*L$|$f(x)$|
+|Level|$2*H$|$12*L$|$f(x)$|
 |:-:|:-:|:-:|:-:|
-1-2|4|48|52
-3-5|6|48|54
-6-7|10|48|58
-8-9|16|72|88
-10-12|26|120|146
-13-14|42|192|234
-15-17|68|312|360
-18-19|110|504|614
-20|178|816|994
+1-2|4|24|28
+3-5|6|24|30
+6-7|10|24|34
+8-9|16|36|52
+10-12|26|60|86
+13-14|42|96|138
+15-17|68|156|224
+18-19|110|252|362
+20|178|408|586
 
 #### How many SP will the player regain from short rests
 Now we have figured out our ideal per long rest SP quotas, we can get a better estimate for the amount of SP to regenerate per short rest. Again, numbers from the our ass, trying to get close to the low tier spell budget at the higher end of the range so high tier spells cut into the players reserves, and low tier spells can be cast without feeling like your dipping into those reserves.
 
-~~If we use $1*level$ then we get $8*6=48$ SP from rests at level 1, and $80*6=480$ SP from rests at level 20. These numbers give us some pretty decent starting points, the ideal max SP goes from 52 at level 1 to 16. ~~
+~~If we use $1*level$ then we get $8*6=48$ SP from rests at level 1, and $80*6=480$ SP from rests at level 20. These numbers give us some pretty decent starting points, the ideal max SP goes from 52 at level 1 to 16.~~
 > Tried so hard to scale by a multiple of character level but it just does not math, either your are over budget by 300 SP until level 15 or underbudget by 700 at higher levels
 
-To do this we can make some more of those tables that we love so much! at lower levels we shoot a little low on the SP regeneration, saying players get 8 SP on a short rest means that our ideal max SP is 4, which allows 2 level one spells which means just about nothing. You will notice that as level goes up the ideal max will fluxuate up and down, this is intentional, we will get into exactly why after this, but just keep that in mind
+~~To do this we can make some more of those tables that we love so much! at lower levels we shoot a little low on the SP regeneration, saying players get 8 SP on a short rest means that our ideal max SP is 4, which allows 2 level one spells which means just about nothing. You will notice that as level goes up the ideal max will fluxuate up and down, this is intentional, we will get into exactly why after this, but just keep that in mind~~
+> rewritting this entirely to better convey how we are choosing numbers
 
-Our new ideal SP by level looks a little like this
-|level|SP/short rest|SP/6 short rests|ideal Max SP|
-|:---:|:-----------:|:--------------:|:-----------:|
-|1|4|24|28|
-|2|5|30|22|
-|3|5|30|24|
-|4|6|36|18|
-|5|7|42|12|
-|6|7|42|16|
-|7|8|48|10|
-|8|10|60|28|
-|9|12|72|16|
-|10|15|90|56|
-|11|18|108|38|
-|12|20|120|26|
-|13|26|156|78|
-|14|32|192|42|
-|15|42|252|108|
-|16|52|312|68|
-|17|62|372|242|
-|18|73|438|176|
-|19|84|504|110|
-|20|100|600|394|
+To properly choose an SP/short rest figure we can't really use an equation (well, strictly speaking we can) because our ideal SP grows exponentially, and any equation on either side makes the players have to recalculate the entire thing every level. So at best we can use one linear equation (add $x$ to this number every level up). During gameplay the entire benefit of equations is that you can level infinitely, so doing that has no point. In practice, I have to make a module for foundry and making all values of SP a function of character level is much easier to program than hard coding everything. It also allows anyone who does want to go over level 20 to start using that equation, and ideally the difficicult equation will be the one that only needs to be solved by one person (Knowing most tables this one will fall on the GM 90% of the time so sorry in advance to all the GMs), and we have an easy linear equation for the one each player needs to solve on their own. 
+
+So which equation only needs to be done once per levelup and which one needs to be done multiple times? The SP per level will be more subject to change between each class than the SP/short rest, so in most cases the SP/short rest will be the same for everyone, while max SP is different between each class. 
+
+Now onto making those equations, the first one we need is how much SP increases each level, we will need this as part of the other equation to adjust for the maxSP pool the player will start with on a long rest. To figure this out, first we ask how much max SP do we want players to have at level 1, and how much for level 20. Say 10sp for level 1 and 200 for 20, that is a difference of 190 across 19 additions, or 10sp for each level. I wasn't thinking about that when chosing that but it works perfectly.
+
+This means a characters max SP can be expressed as $ 10 * level $ so we will subtract that from our answer when we get to SP regeneration
+
+The annoying equation is as follows, and will be explained in sections after the full definition
+
+$
+\begin{aligned} \\
+    &\text{let } x = \text{Character level} \\
+    &\text{let H} = \begin{cases}
+        2  _1 & \text{if } 1  \leq x \leq 2  _1 \\
+        3  _2 & \text{if } 3  \leq x \leq 5  _2 \\
+        5  _3 & \text{if } 6  \leq x \leq 7  _3 \\
+        8  _4 & \text{if } 8  \leq x \leq 9  _4 \\
+        13 _5 & \text{if } 10 \leq x \leq 12 _5 \\
+        21 _6 & \text{if } 13 \leq x \leq 14 _6 \\
+        34 _7 & \text{if } 15 \leq x \leq 17 _7 \\
+        55 _8 & \text{if } 18 \leq x \leq 19 _8 \\
+        89 _9 & \text{if } 20 \leq x         _9 
+    \end{cases}\\
+    &\text{let L} = \begin{cases}
+        2  _1 & \text{if } 1  \leq x \leq 7  _1 \\
+        3  _2 & \text{if } 8  \leq x \leq 9  _2 \\
+        5  _3 & \text{if } 10 \leq x \leq 12 _3 \\
+        8  _4 & \text{if } 13 \leq x \leq 14 _4 \\
+        13 _5 & \text{if } 15 \leq x \leq 17 _5 \\
+        21 _6 & \text{if } 18 \leq x \leq 19 _6 \\
+        34 _7 & \text{if } 20 \leq x         _7
+    \end{cases}\\
+    &S=\frac{(12L+2H-10x)}{3}
+\end{aligned}
+$