How to generate random waves for a bullet hell game that feel balanced and natural

My game consists of ‘waves’ of objects called ‘spawners’, which once every certain amount of time (their firetime), move to a new place on the screen and spawn an enemy. Each wave has 4 important properties:
1: the amount of spawners the wave has
2: the interval of time between creating new spawners
3: the total length of the wave
4: the types of spawners that can appear in a wave (represented as a std::vector<std::pair<std::string, int>> where the string is the spawner name, and the integer is its spawn weight.

The game works by picking a random spawner from the possible spawner types (with a weighted rng) every new spawner interval. Currently waves are set and are loaded from a file at runtime.

My problem is that I cannot find a good way to randomly generate waves that feel balanced and natural. Currently, I am trying to generate waves based on a difficulty value, mostly using weighted random number generation. However, this does not produce balanced waves that correspond well to the target difficulty. Even after trying several different techniques, I am unable to get a system that generates waves that fell balanced and natural (like the ones hand made).

Is there any way to generate waves that feel natural, based off of the difficulty value? How should I approach this problem?

Also if its of any help, each spawner also defines its own difficulty value.

“Searching and sorting” algorithm to find the natural logarithim of a number?

Yeah, this is for a homework assignment, but I hope that you’ll humor me anyway. I am asked to design an algorithm that finds the natural logarithm of a number. This would be straightforward, but I’m not allowed to use strategies that involve integration, bit manipulation, approximation formulas, or Taylor Series. Instead, we’re asked to generate an array of values where at least one of them will be the answer, and then to find the correct value in that array. There must be some kind of established algorithm for this, but this and other resources haven’t been helpful. I’m stuck. What should I be thinking about here?

Placing Critters in the appropriate place in the Natural Order Hierarchy in Mouse Guard

I am currently developing a campaign setting using the rules and lore for Mouse Guard. The setting will be similar but not in the same geographical location (Northern Michigan) that the original was written in. To that end I will be bringing in additional animals and developing their stats. I asked a question here that references how to balance the Nature scores and encounter abilities of the creatures but will also need to determine how best to place new creatures in the Natural Order table.

In MG mice can only directly attack/kill creatures that are up to two levels higher than they are in the hierarchy. Where to place a new creature in this table could indicate mechanically whether they would need extraordinary means to interact with that creature during an encounter. Balancing this would be an important part of creating a new creature in a custom Mouse Guard campaign.

I’m looking for expertise in balancing new creatures and placing them within the Natural Order hierarchy when introducing them into a Mouse Guard campaign.

I’m looking for answers using the “Good Subjective” standard and experience on how specifically balancing was done and determining where to place the creatures in the Natural Order hierarchy.

How would you optimize a multiclass druid/monk for DPR assuming Natural Weapons are considered Unarmed Strikes?

Assuming that the Unarmored Defense, and Flurry of Blows features both work with the Druid’s Wild Shape feature, what is the optimal DPR build for a Druid/Monk multiclass, and what are the notable break points for this particular multiclass?

You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so.
PHB, page 67

Also assume that Natural Weapons are considered Unarmed Strikes, as per the the description of Alter Self:

Natural Weapons. You grow claws, fangs, spines, horns, or a different natural weapon of your choice. Your unarmed strikes deal 1d6 bludgeoning, piercing, or slashing damage, as appropriate to the natural weapon you chose, and you are proficient with your unarmed strikes.
PHB, page 212

Given access to any appropriate Beast, as well as having the option to choose either:
a) Multi-attack from the Beast entry, or
b) Extra Attack from the appropriate Monk level
What is the highest DPR possible?

Other assumptions:

  • 27 Point Buy
  • Official D&D Content only, excluding Unearthed Arcana

Can an Eldritch Knight bond with a natural weapon?

Let’s say an Eldritch Knight (lvl 8) casts Alter Self and gains claws as natural weapons, could he then create a Weapon Bond with his claws?

Relevant Rules:

MELEE AND RANGED ATTACKS The most common actions that a monster will take in combat are melee and ranged attacks. These can be spell attacks or weapon attacks where “weapon” might be a manufactured item or a natural weapon such as a claw or tail spike. (MM p10)

Weapon Bond At 3rd level, you learn a ritual that creates a magical bond between yourself and one weapon… Once you have bonded a weapon to yourself, you can’t be disarmed of that weapon unless you are incapacitated. If it is on the same plane of existence, you can summon that weapon as a bonus action on your turn, causing it to teleport instantly to your hand. (PHB p75)

Alter Self; Natural Weapons. You grow claws, fangs, spines, horns, or a different natural weapon of your choice. Your unarmed strikes deal 1d6 bludgeoning, piercing, or slashing damage, as appropriate to the natural weapon you chose, and you are proficient with your unarmed strikes. Finally, the natural weapon is magic and you have a +1 bonus to the attack and damage rolls you make using it. (PHB 211)

How does the Natural Weapons option of the Alter Self spell affect a Longtooth Shifter’s Shifting Feature?

I play an Eldritch Knight Longtooth Shifter who’s about to get 2nd level spells. I was looking at Alter Self (which I realize I can’t get until a level later anyway because of its school) and I was curious about how it interacted with my Shifting feature.

[Longtooth] Shifting Feature. While shifted, you can use your elongated fangs to make an unarmed strike as a bonus action. If you hit with your fangs, you deal piercing damage equal to 1d6 + your Strength modifier, instead of the bludgeoning damage normal for an unarmed strike.

So my shifting feature grants me an unarmed strike attack as a bonus action using elongated fangs. Looking at Alter Self:

Natural Weapons. You grow claws, fangs, spines, horns, or a different natural weapon of your choice. Your unarmed strikes deal 1d6 bludgeoning, piercing, or slashing damage, as appropriate to the natural weapon you chose, and you are proficient with your unarmed strikes. Finally, the natural weapon is magic and you have a +1 bonus to the attack and damage rolls you make using it.

If I cast this spell using this option and choose to “grow fangs” while shifted, does the bonus unarmed strike have a +1 bonus because of Alter Self?

Pathfinder Natural Weapon Damage by Size Inconsistency

Source: https://www.d20pfsrd.com/bestiary/rules-for-monsters/universal-monster-rules/#Natural_Attacks

Take for example a bite attack. According to the table, a small bite does 1d4 damage, medium 1d6, large 1d8, huge 2d6, garg. 2d8, and col. 4d6. Presumably it’d continue 4d8, 8d6, etc.

However, if you look at the FAQ entry right below it, it says that as you increase the damage, the progression would be 1d4, 1d6, 1d8, 2d6, 3d6, 4d6, 6d6, 8d6, 12d6, etc.

So what is this supposed to mean? Does it mean for example that a gargantuan entity would do 2d8 damage with a bite attack, but a huge one enlarged to gargantuan would do 3d6?

This seems it might be a duplicate of Is there an error in the universal monster rules for table:natural attacks by size? That question is 2 years old. Have we gotten any new information since then? Also, the newer chart is listed as a FAQ instead of an errata, which implies that the old chart might not be a mistake?

Razborov’s natural proof barrier applies to which complexity classes?

From what I understand, Razborov’s natural proof barrier applies to certain complexity classes such as $ P$ and $ P/poly$ . Suppose we intend to separate any two complexity classes with a natural proof, how do we know if the barrier applies to this case or not?

I know this question is vague, but this is because I do not understand what specifically about a lower bound proof against $ P$ and $ P/poly$ but not $ AC^0$ invalidates the existence of a pseudorandom generator. Any intuition you could provide?

are natural weapons considered weapons for the purpose of spell components?

Are “natural weapons” considered weapons for spells that require weapons as a material component? I have found a few questions that touch on the specific topic but together they are inconclusive.

Some related questions:

What are natural weapons?

Do natural weapons work with the Booming Blade spell?

Can natural weapons be simple or martial weapons?