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I'm unclear on what a wizard's “spells known” means. Namely, are they the spells they can prepare, or the spells in their spellbook?

This is probably really obvious, but I'm just not figuring it out. Thanks!

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    \$\begingroup\$ Can you give us an example of a situation where the spells a wizard can prepare and the spells they have in their spellbook are different? I'm not saying that's not possible—it can easily happen—but if you provide an example situation we'll be able to see more clearly exactly what difficulty you're focused on and stumbling over. Thanks! \$\endgroup\$ Jun 13 '16 at 20:21
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"Spells Known" refers to the spells the wizard currently has written in their spellbook.

From the PHB Page 114:

Spellbook
At 1st level you have a spellbook containing six 1st-level wizard spells of your choice.

...

Learning Spells of 1st Level and Higher
Each time you gain you gain a wizard level you can add two wizard spells must be of a level for which you have spell slots, as shown on the Wizard table. On your adventures you might find other spells you can add to your spellbook.

Okay so let's assume you're a level 2 Wizard who hasn't learned any external spells. Your Spells Known would total 8.

Now as to spells prepared you can prepare Int Mod + Wizard Level # of spells each day. So let's say you have an intelligence of 16 (mod of +3), at level 2 you'd be able to prepare 5 spells from the list of Spells Known.

As you adventure you can come across other spells and copy them down into your spellbook. Your biggest concern is not to lose your spellbook, or to make sure you have a copy of it. Check the Your Spellbook sidebar on page 114 for doing just that.

TL;DR

Spells Known are spells in your spellbook. From those spells, you prepare a number of spells to cast each day equal to (Intelligence Modifier + Wizard Level). These spells you've selected are your Prepared Spells.

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Spells you know refers to a couple of things:

1) The spells which you have written in your spellbooks. Notice the plural. The PHB states:

many wizards keep backup spellbooks in a safe place

The only limitation is the time and money it takes to transcribe the spells.

2) The spells you currently have memorized.

If you lose your spellbook, you can use the same procedure to transcribe the spells that you have prepared into a new spell book

If your spellbooks are destroyed, you can re-transcribe a new spell book from spells that you have in memory. Note that this is almost certainly a small subset of what was in the book, and requires you to not cast anything until you have had a chance to write it down.

Spells you can prepare refers to ones you can memorize and cast. This requires the spell book and the spells known that are written in it. Note that a frugal wizard could have a master spellbook at home under lock and key, and have a cheaper "traveling spellbook" to carry with them, containing only the spells they will likely need while adventuring. It would also keep enemies from gaining their entire spell list should they be captured and looted. Thus "spells you can prepare" does not always equal "spells known". The former is situational.

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