Magic systems

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Slot-based
Spells must be prepared in advance, after which it may be cast at any time. In many systems, once a spell is cast, it is consumed; to cast a spell twice, it must be prepared twice.
Typically the maximum number of spells that can be prepared is finite, either with an explicit cap ('Slots'), or by requiring a finite resource (i.e. material components).
Variations include: more powerful spells require more slots/resources; D&D-style leveled slots; prerequisite spells that implicitly consume extra slots for powerful spells.
Mana-Based
Spells consume spell points ('Mana') when cast, but spellcasting is otherwise unrestricted.
Variations include: 'Aspected Mana' (fire mana, water mana, offensive mana, defensive mana, etc.); spells consuming things other than mana (hit points, material components, food/nutrition, etc.); spells consume more than one type of thing/aspect of mana (e.g. 3 fire mana and a stick of charcoal).
Chance-Based
Spellcasting is essentially free, but has a large chance of failure, especially for high-level spells.
A variant is to have the chance of failure be affected by some combination stats, skill, or environment.

It is quite common to combine more than one of these methods.

A Mana/Chance system might allow memorization of any number of spells, with the drawback that they cost mana and may fail. NetHack does this.

A Slot/Mana/Chance system might allow you to memorize a short list of spells, which then may be cast as above. Crawl does this.

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