Is it possible to ask Mathematica to give all the roots of the given function?

I have this equation

$ $ 36 \cos \frac{3 x}{4} \cos \frac{27 x}{20} \left(\cos \frac{3 x}{5} +2 \cos \frac{21 x}{10} \right)=0 $ $

Is it possible to ask Mathematica to give all the roots of the function on the domain $ 0<x<4\pi$ ? Preferably as a rational multiple of $ \pi$ ? (the plot of the function is attached)

36 Cos[(3 x)/4] Cos[(27 x)/20] (Cos[(3 x)/5] + 2 Cos[(21 x)/10]) == 0  

enter image description here

How do I give the right amount of treasure without going overboard, like my last DM did?

I am trying to find a good model for how much treasure and money to give my players in order not to make items too difficult to buy, or too cheap to buy.

In my last game, which ran for 6 months, our DM ended up so freely giving us gold that everything was trivial for us. Our characters at level 6 were so rich, we could afford any comfort and travel as kings. This was not because we found a treasure hoard, but because our DM had accidentally been giving us too much treasure for our encounters.

For example, every time we would fight something, they would reward us with 1-2 Platinum, sometimes more. They also used in-game currency for out of game creativity. If you made them laugh, you got another Platinum or like 100 Gold.

Nearing the end of the campaign, everything was so cheap that it wasn’t fun anymore to even buy anything or save up for fancy lodgings. We could afford hundreds of potions, countless trips to the local temple, resurrections…etc. There was no threat of dying, no challenge in earning, and the game lost steam fast.

I am not good with math. I am nowhere near an economist, but when I run my campaign I don’t want to end up with the same problem.

Are there any charts or rules of thumb for making a balanced, long-term economy in D&D 5th?

How do I assign treasure in a way that doesn’t inflate to worthlessness down the line?

I am looking for charts, guides or tables ideally. I would really appreciate the help.

Does the Faerie Fire spell give advantage on attacks against invisible creatures?

The description of the faerie fire spell states:

Any attack roll against an affected creature or object has advantage if the attacker can see it, and the affected creature or object can’t benefit from being invisible.

As I read it, there are two ways to interpret this. The first is that being held unseen is a benefit of being invisible, and therefore the spell removes that benefit. Since the invisible creature is then visible, you have advantage against it.

The other interpretation is that the order of the sentence matters; first, check if you can see them, and you have advantage if you can. Then, strip them of the benefits of invisibility. In this case, you would have a regular attack roll against the creature, without disadvantage from being invisible nor advantage from Faerie Fire.

What interpretation of the rule aligns with the intention of the faerie fire spell?

Which Spheres of Might talents give a Vector attacks through Kinetic Overload?

The vector symbiat archetype, from Champions of the Spheres, has the ability Kinetic Overload, which reads, in part:

When a Brute, Scoundrel, or Wrestling talent would grant the vector an attack against a target within his telekinesis range as the result of performing a maneuver [emphasis mine], he may choose to make a ranged attack roll to pummel his target in place of the attack, dealing 1d6 + his casting ability modifier bludgeoning damage. This damage increases by 1d6 at 3rd level and every two levels thereafter and counts as magic.

My question is which martial talents from these three spheres actually grant these attacks? Some seem to obviously grant the attack (Scoundrel’s Mug, Wrestling’s Chink in the Armor), and others probably not (Wrestling’s Slip and Strike), but what about an ability such as Wrestling’s Piledriver?

When you successfully maintain a grapple against a creature you have already pinned, instead of one of the options normally available when you maintain a grapple, you may lift them over your head and smash them into the ground, automatically threatening a critical hit with your unarmed strike and rolling to confirm as normal. If the critical threat confirms successfully, the creature must make a successful Fortitude save or be staggered for 1 round. Use of this talent breaks your grapple on the creature. At +10 base attack bonus, the target is dazed for 1 round instead.

Does this fulfill the twin parts of Kinetic Overload, namely, "making an attack" (it’s automatically threatening a critical hit; no attack roll is being made, but a confirmation roll is called for), and "as the result of performing a maneuver" (it’s an option upon maintaining a grapple)? Does Kinetic Overload’s option to make the attack with telekinetic force override Piledriver’s requirement that the attack be made with an unarmed strike?

Piledriver’s the most confusing example I’ve found in my readings, but I’d like to see either a specific set of tests (does it have to be an attack roll, does it have to allow any weapon to be used with it, etc.) or a full list of all talents that grant an attack that Kinetic Overload can replace this way (probably no more than a dozen unless it’s a lot more generally applicable than I’m understanding it to be). I’m also working under the understanding that the attacks made as part of shove, marked target, and snag do not qualify, since they’re handled in their own paragraph of Kinetic Overload.

How can I calculate the optimal amount of attack bonus to give up when using Power Attack?

The Power Attack feat lets you sacrifice attack bonus (AB) to increase your damage.

It’s easy to see that you don’t always want to give up the same amount of AB, and that the optimal amount (assuming you’re trying to maximize your expected damage output) depends on multiple factors. Consider the following two degenerate cases where it’s easy to see that the optimal choice differs:

  1. If your opponent’s AC is so much higher than your AB that you will only ever hit them by rolling a natural 20, then you clearly want to sacrifice the maximum possible amount of AB, since it won’t affect your chance to hit at all (5% in any case), and maximizing your Power Attack damage will result in more damage if you do roll the natural 20.
  2. If your foolish DM has granted you the legendary +Graham’s Number Sword of Munchkinry at level 1, then you clearly don’t want to use Power Attack at all (unless you’ll only miss on a natural 1), because your base damage is so high that even a 5% decrease in your chance to hit will utterly dwarf any piddly damage you get from sacrificing your 1 BAB.

In between these silly cases, though, I’m not sure how to determine the best amount of AB to sacrifice when using Power Attack. It’s not even clear to me what information I need to do so, though I think it includes some or all of the following:

  • The attacker’s AB
  • The defender’s AC
  • The attacker’s BAB (because it’s the maximum amount of AB they can subtract)
  • The number of attacks the attacker is making (if they’re making a full attack)
  • The amount of damage the attack(s) will do on hit
  • How much damage the attacker gains per point of sacrificed AB (e.g., 2 points when using a 2-handed weapon instead of 1 point for a 1-handed weapon)
  • Whether the attack could crit, and its crit stats (range/multiplier) if so

Given this sort of info, how can I calculate the amount of AB to sacrifice to Power Attack that maximizes my expected damage output for the round?

A couple notes:

  • I will happily upvote partial solutions (e.g., ones that only apply to a single attack without considering iteratives, or ones that ignore crits for simplicity’s sake)
  • Ignore considerations that require you to know how close to death the defender is. I’m happy with answers that naively maximize the expected value of my damage output against an idealized combat dummy with infinite HP. Accounting for the desire to maximize the probability of dealing lethal damage against low-HP opponents is, I think, too complicated, and beyond the scope of this question.
  • Ignore the Shock Trooper feat for purposes of this question; obviously if you’re giving up AC for damage instead of AB, it’s a risk/reward judgment call, not a case where there’s an objectively optimal value.

Can a familiar give orders to minions of their wizard?

A wizard has a familiar and some other minions. Let’s go with a summoned monster and a non-intelligent undead minion. Can the familiar give commands to these other minions?

Does it matter if the wizard is present? (Like if they get teleported away) Does it matter if the wizard is conscious? Does it matter if the familiar can speak? Does it matter if how the wizard is controlling the undead (create undead vs command undead)?

Any other edge cases one can think of?

What Should I Give Level 20 Players [closed]

I really like giving out loot and spells but what should I give to level 20 players. It would be also really nice if someone should tell me what to give to low level players as well. I do own a copy of the DM’s handbook (and it tells me what to give) but should I give the less, more or the right amount. I’m asking about 5 edition dungeons and dragons.

Does a clay golem’s haste action actually give it more attacks?

A clay golem has an action called Haste that is thematically similar to the spell of the same name but mechanically rather different:

Haste (Recharge 5–6). Until the end of its next turn, the golem magically gains a +2 bonus to its AC, has advantage on Dexterity saving throws, and can use its slam attack as a bonus action.

At first glance, this appears to grant the golem one additional attack (as a bonus action). However, using the ability costs the golem its action, with which it could otherwise make 2 slam attacks. Since the ability lasts until the end of the golem’s next turn, it can attack with its bonus action on both the turn it uses the ability and the following turn, yielding a total of… 2 additional attacks. Thus, it seems that over the course of these 2 turns, the golem will get 4 attacks, regardless of whether it uses its Haste ability or not. The only difference is that with Haste, one of those attacks comes 1 turn later.

So, am I correct in finding that the net effect of the golem’s Haste action is to delay one of its attacks by one turn in return for improved defenses for 1 round, as opposed to actually granting additional attacks? Obviously there are cases where this is unambiguously a benefit, e.g. when the golem can’t get into melee on the current turn, but I suppose I find it a bit surprising that, if anything, the golem’s Haste action causes it to attack more slowly.

What level would you give the players a dancing sword?

I am currently DMing a campaign for level 5 players and one of them wants a dancing sword. I know the weapon is labelled as "very rare" but it doesn’t seem all that more powerful than a cleric with spiritual weapon. My question is when would you give the players an item like that and would you add any +1/+2/+3 modifiers? Note: I know about the recommendations regarding awarding magic items in the books. I just want to know if anyone can justify why it has the "very rare" rating and when you might give it out instead?