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Most every class that can cast a spell has a Spell Save DC, which is typically

DC = 8 + proficiency + [ability modifier]

However, if a monk follows the Way of the Four Elements, they gain access to several abilities that either mimic existing spells with saving throws (Burning Hands) or create new spell-like abilities with saving throws (Fist of Unbroken Air).

However, these spells don't explicitly spell out which ability modifier should be used for the save DC. I have gone with Wisdom in personal games, as it's something that Monks already have a good deal of, but I cannot find anything in the PHB backing this up.

Which ability, according to the rules, should Monks use to get their spell save DC?

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You use your Wisdom to set your spell save DC.

Under the Ki section of p. 78:

Ki save DC: 8 + your proficiency bonus + your Wisdom modifier

All of your abilities that cast spells are Ki abilities, so you're spending Ki points to cast spells. Since they are Ki abilities, they use your Ki ability DC.

For instance, when you spend 2 ki points to cast burning hands, it's DC is 8 + proficiency bonus + Wisdom modifier.

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