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I am DMing the D&D 5e Starter Set campaign, Lost Mine of Phandelver, for a group of my friends. They recently searched the Redbrand Hideout in Tresendar Manor in Phandalin, and searched through the wizard's workshop; among other things, they found some of his notes. The description in the adventure book (p. 25) states:

Books and Notes. Iarno is trying to master the art of brewing potions and concocting alchemical mixtures. The books and notes scattered around the room are basic texts on alchemy. Any character proficient in Arcana can see that Iarno's apparatus appears to be set up to brew potions of invisibility—not that he has succeeded so far.

In response to this, the Wizard character in the party asked if he could now make Potions of Invisibility. I initially just made up a reason why he wouldn't be able to do so (since I wasn't sure at the time – and upon rereading, it also looks like I missed the fact that the books themselves are just basic texts on alchemy, not instructions on making the potions), but he wasn't satisfied with that, and I was also personally curious as to how he would go about crafting such an item.

What would be required in order for a character to craft Potions of Invisibility?

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The Dungeon Master's Guide lists "Crafting a Magic Item" as an option under "More Downtime Activities" (p. 128–129):

Magic items are the DM's purview, so you decide how they fall into the party's possession. As an option, you can allow player characters to craft magic items.

The creation of a magic item is a lengthy, expensive task. To start, a character must have a formula that describes the construction of the item. The character must also be a spellcaster with spell slots and must be able to cast any spells that the item can produce. Moreover, the character must meet a level minimum determined by the item's rarity, as shown in the Crafting Magic Items table.

The DMG then provides an example, and suggests that the DM may also require that certain items need special materials or locations in order to be made, such as alchemist's supplies to brew a certain potion.

The DMG continues:

An item has a creation cost specified in the Crafting Magic Items table (half that cost for a consumable, such as a potion or scroll). A character engaged in the crafting of a magic item makes progress in 25 gp increments, spending that amount for each day of work until the total cost is paid. The character is assumed to work for 8 hours each of those days. Thus, creating an uncommon magic item takes 20 days and 500 gp.

If a spell will be produced by the item being created, the creator must expend one spell slot of the spell's level for each day of the creation process. The spell's material components must also be at hand throughout the process.

(The halved cost for consumables is a correction stated in the DMG errata.)

The Crafting Magic Items table is as follows:

\begin{array}{l | l | l} \text{Item Rarity} & \text{Creation Cost (gp)} & \text{Minimum Level} \\ \hline \text{Common} & 100 & 3rd \\ \text{Uncommon} & 500 & 3rd \\ \text{Rare} & 5,000 & 6th \\ \text{Very Rare} & 50,000 & 11th \\ \text{Legendary} & 500,000 & 17th \\ \end{array}

According to the Magic Items by Rarity list on the Wizards of the Coast website, a Potion of Invisibility is considered "Very Rare", so the creation cost would be 25,000 gp (halved from 50,000 because it's a consumable) and the minimum level needed to craft it would be level 11. A Potion of Invisibility would also need to be crafted by someone who has a 3rd-level spell slot and knows Invisibility (only the Bard, Sorcerer, Warlock and Wizard have these on their spell lists, but other classes can get access to them).

Given that creation cost, it would take a single character of the requisite level and class 1,000 days to craft one Potion of Invisibility, spending 25,000 gp to do so.

Note that the DMG does include rules on assisting in making a magic item:

Multiple characters can combine their efforts to create a magic item if each of them meets the level prerequisite. Each character can contribute Spells, spell slots, and components, as long as everyone participates during the entire crafting process. Each character can contribute 25 gp worth of effort for each day spent helping to craft the item.

So, for instance, if the character is in a party with 3 others of 11th level or higher, they can help contribute to the cost in terms of time and money (and spell slots and components if relevant). With a Wizard crafting the potion and 3 other characters assisting, the total creation cost (25,000 gp) would remain unchanged but the process would only take one-fourth as long: 250 days.

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